Cabozantinib-S-Malate in Treating Younger Patients With Recurrent, Refractory, or Newly Diagnosed Sarcomas, Wilms Tumor, or Other Rare Tumors

Inclusion Criteria:

* Upper age limit of =\< 18 years of age for medullary thyroid carcinoma (MTC), renal cell carcinoma (RCC) and hepatocellular carcinoma (HCC) =\< 30 years for all other diagnoses
* Patients must have a body surface area \>= 0.35 m\^2
* Patients must have recurrent or refractory disease, or newly diagnosed disease with no known curative therapy or therapy proven to prolong survival with an acceptable quality of life; patients must have had histologic verification of one of the malignancies listed below at original diagnosis or at relapse:

  * Ewing sarcoma
  * Rhabdomyosarcoma (RMS)
  * Non-rhabdomyosarcoma soft tissue sarcomas (STS) including microphthalmia transcription factor associated STS (alveolar soft part sarcoma \[ASPS\] and clear cell sarcoma \[CCS\])
  * Osteosarcoma
  * Wilms tumor
  * Rare tumors

    * Medullary thyroid carcinoma (MTC)
    * Renal cell carcinoma (RCC)
    * Hepatocellular carcinoma (HCC)
    * Hepatoblastoma
    * Adrenal coertex carcinoma
    * Pediatric solid tumors (including central nervous system \[CNS\] tumors) with known molecular alterations in the targets of XL184 (i.e., MET amplification, overexpression, activating mutation, MET translocation, MET exon skipping mutations, activating RET mutations, RET rearrangement, overexpression or activation of AXL); documentation of the alteration from a Clinical Laboratory Improvement Act (CLIA) certified laboratory will be required

      * Note: Documentation of any known tumor molecular alterations and RET mutation status for patients with MTC (germline) must be uploaded via the RAVE system
* Patients must have radiographically measurable disease; measurable disease is defined as the presence of at least one lesion on magnetic resonance imaging (MRI) or computed tomography (CT) scan that can be accurately measured with the longest diameter a minimum of 10 mm in at least one dimension (CT scan slice thickness no greater than 5 mm)

  * Note: The following do NOT qualify as measurable disease:

    * Malignant fluid collections (e.g., ascites, pleural effusions)
    * Bone marrow infiltration
    * Lesions only detected by nuclear medicine studies (e.g., bone, gallium or positron emission tomography \[PET\] scans)
    * Elevated tumor markers in plasma or cerebrospinal fluid (CSF)
    * Previously radiated lesions that have not demonstrated clear progression post radiation
    * Leptomeningeal lesions that do not meet the measurement parameters noted above
* Patients must have a Lansky or Karnofsky performance status score of \>= 50, corresponding to Eastern Cooperative Oncology Group (ECOG) categories 0, 1 or 2; use Karnofsky for patients \> 16 years of age and Lansky for patients =\< 16 years of age; patients who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score
* Patients must have fully recovered from the acute toxic effects of all prior chemotherapy, immunotherapy, or radiotherapy prior to entering this study
* Patients with solid tumors must not have received myelosuppressive chemotherapy within 3 weeks of enrollment onto this study (6 weeks if prior nitrosourea)
* At least 7 days must have elapsed since the completion of therapy with a growth factor. At least 14 days must have elapsed after receiving pegfilgrastim
* Anti-cancer agents not known to be myelosuppressive (e.g. not associated with reduced platelet or absolute neutrophil count \[ANC\] counts): \>= 7 days after the last dose of agent
* Antibodies: \>= 21 days must have elapsed from infusion of last dose of antibody, and toxicity related to prior antibody therapy must be recovered to grade =\< 1
* \>= 2 weeks must have elapsed since local palliative radiation therapy (XRT) (small port); \>= 6 weeks must have elapsed since treatment with therapeutic doses of M-Iodobenzylguanidine (MIBG); \>= 3 months must have elapsed if prior craniospinal XRT was received, if \>= 50% of the pelvis was irradiated, or if total-body irradiation (TBI) was received; \>= 6 weeks must have elapsed if other substantial bone marrow irradiation was given

  * Subjects should not have any clinically relevant ongoing complications from prior radiation therapy (i.e., radiation esophagitis or other inflammation of the viscera)
* No evidence of active graft versus (vs.) host disease and \>= 2 months must have elapsed since transplant
* Not previously received XL184 or another MET/HGF inhibitor (tivantinib or crizotinib); there are no limits on number of prior therapeutic regimens; patients who have been treated with prior VEGF pathway, or RET inhibitors (except XL184) may be eligible
* Peripheral absolute neutrophil count (ANC) \>= 1000/uL for patients with solid tumors without bone marrow involvement
* Platelet count \>= 100,000/uL (transfusion independent, defined as not receiving platelet transfusions within a 7 day period prior to enrollment) for patients with solid tumors without bone marrow involvement
* Hemoglobin \>= 8.0 g/dL (may receive red blood cell \[RBC\] transfusions) for patients with solid tumors without bone marrow involvement
* Peripheral absolute neutrophil count (ANC) \>= 750/uL for patients with solid tumors and known bone marrow metastatic disease
* Platelet count \>= 50,000/uL for patients with solid tumors and known bone marrow metastatic disease
* Hemoglobin \>= 8.0 g/dL for patients with solid tumors and known bone marrow metastatic disease
* Transfusions are permitted to meet both the platelet and hemoglobin criteria for patients with known bone marrow metastatic disease; patients must not be known to be refractory to red blood cell or platelet transfusions
* Creatinine clearance or radioisotope glomerular filtration rate (GFR) \>= 70 mL/min/1.73 m\^2 or a serum creatinine based on age/gender as follows:

  * 2 to \< 6 years of age

    * Male and female: 0.8 (maximum serum creatinine \[mg/dL\])
  * 6 to \< 10 years of age

    * Male and female: 1 (maximum serum creatinine \[mg/dL\])
  * 10 to \< 13 years of age

    * Male and female: 1.2 (maximum serum creatinine \[mg/dL\])
  * 13 to \< 16 years of age

    * Male 1.5 (maximum serum creatinine \[mg/dL\])
    * Female: 1.4 (maximum serum creatinine \[mg/dL\])
  * \>= 16 years of age

    * Male: 1.7 (maximum serum creatinine \[mg/dL\])
    * Female: 1.4 (maximum serum creatinine \[mg/dL\])
* Urine protein: =\< 30 mg/dl in urinalysis or =\< 1+ on dipstick, unless quantitative protein is \< 1000 mg in a 24 hour (h) urine sample
* Total bilirubin =\< 1.5 x upper limit of normal (ULN) for age
* Serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 135 U/L (3 x ULN) (for the purpose of this study, the ULN for SGPT is 45 U/L)
* Serum albumin \>= 2.8 g/dL
* No history of congenital prolonged corrected QT (QTc) syndrome, New York Heart Association (NYHA) class III or IV congestive heart failure (CHF)
* No clinically significant cardiac arrhythmias, stroke or myocardial infarction within 6 months prior to enrollment
* QTc =\< 480 msec; Note: Patients with grade 1 prolonged QTc (450- 480 msec) at the time of study enrollment should have correctable causes of prolonged QTc addressed if possible (i.e., electrolytes, medications)
* Patients with a known seizure disorder who are receiving non-enzyme inducing anticonvulsants and have well-controlled seizures may be enrolled
* CNS toxicity =\< grade 2 with the exception of decreased tendon reflex (DTR); any grade of DTR is eligible
* A blood pressure (BP) =\< the 95th percentile for age, height, and gender for pediatric patients \< 18 years old and =\< 140/90 mmHg for patients \>= 18 years old; patients should not be receiving medication for treatment of hypertension (except patients with Wilms tumor and RCC who may be eligible if on stable doses of no more than one anti-hypertensive medication with a baseline BP =\< ULN for pediatric patients and =\< 140/90 for adult patients); please note that 3 serial blood pressures should be obtained and averaged to determine baseline BP
* International normalized ratio (INR) =\< 1.5
* Serum amylase =\< 1.5 x ULN
* Serum lipase =\< 1.5 x ULN

Exclusion Criteria:

* Pregnant or breast-feeding women will not be entered on this study due to risks of fetal and teratogenic adverse events as seen in animal/human studies; pregnancy tests must be obtained in girls who are post-menarchal; males or females of reproductive potential may not participate unless they have agreed to use two methods of birth control- a medically accepted barrier method of contraceptive method (e.g., male or female condom) and a second effective method of birth control-during protocol therapy and for at least 4 months after the last dose of XL184; abstinence is an acceptable method of birth control
* Growth factors that support platelet or white cell number or function must not have been administered within the 7 days prior to enrollment (14 days if pegfilgrastim)
* Patients requiring corticosteroids who have not been on a stable or decreasing dose of corticosteroid for the 7 days prior to enrollment are not eligible; if used to modify immune adverse events related to prior therapy, \>= 14 days must have elapsed since last dose of corticosteroid
* Previous treatment with XL184 (cabozantinib) or another MET/HGF inhibitor (tivantinib, crizotinib)
* Patients who are currently receiving another investigational drug are not eligible
* Patients who are currently receiving other anti-cancer agents are not eligible
* Patients who are receiving cyclosporine, tacrolimus or other agents to prevent either graft-versus-host disease post bone marrow transplant or organ rejection post-transplant are not eligible for this trial
* Patients must not be receiving any of the following potent CYP3A4 inducers or inhibitors: erythromycin, clarithromycin, ketoconazole, azithromycin, itraconazole, grapefruit juice or St. John's wort
* Concomitant anticoagulation with oral anticoagulants (e.g., warfarin, direct thrombin, and Factor Xa inhibitors) or platelet inhibitors (e.g., clopidogrel) are prohibited

  * Note: Low-dose aspirin for cardioprotection (per local applicable guidelines) and low dose, low molecular weight heparins (LMWH) are permitted; anticoagulation with therapeutic doses of LMWH is allowed in subjects without radiographic evidence of brain metastasis, who are on a stable dose of LMWH for at least 6 weeks before first dose of study treatment, and who have had no complications from a thromboembolic event or the anticoagulation regimen
* Patients must not have received enzyme-inducing anticonvulsants within 14 days prior to enrollment
* Patients who are receiving drugs that prolong QTc are not eligible
* Patients who are unable to swallow intact tablets are not eligible
* Patients who have an uncontrolled infection are not eligible
* Patients who in the opinion of the investigator may not be able to comply with the safety monitoring requirements of the study are not eligible
* Patients with active bleeding are not eligible; specifically, no clinically significant gastrointestinal (GI) bleeding, GI perforation, intra-abdominal abscess or fistula for 6 months prior to enrollment, no hemoptysis or other signs of pulmonary hemorrhage for 3 months prior to enrollment; patients with evidence of an acute intracranial or intratumoral hemorrhage on CT or MRI are not eligible (patients with evidence of resolving hemorrhage will be eligible); in patients with CNS tumors, an MRI with ECHO gradient sequences would be required to exclude presence of petechial hemorrhages
* Patients who have had or are planning to have the following invasive procedures are not eligible:

  * Major surgical procedure, laparoscopic procedure, or open biopsy within 28 days prior to enrollment
  * Central line placement or subcutaneous port placement is not considered major surgery but must be placed at least 3 days prior to enrollment for external lines (e.g., Hickman or Broviac catheter, peripherally inserted central catheter \[PICC\]) and at least 7 days prior to enrollment for a subcutaneous port
  * Core biopsy within 7 days prior to enrollment
  * Fine needle aspirate within 7 days prior to enrollment
  * Surgical or other wounds must be adequately healed prior to enrollment
  * NOTE: For purposes of this study, bone marrow aspirate and biopsy are not considered surgical procedures and therefore are permitted within 14 days prior to start of protocol therapy
* Patients who have had significant traumatic injury within 28 days prior to enrollment are not eligible
* Patients with any medical or surgical conditions that would interfere with gastrointestinal absorption of the study drug are not eligible