Inclusion Criteria:

* Patients must have histologically or cytologically confirmed papillary, follicular, or Hurthle cell carcinoma (cohort A) or medullary thyroid carcinoma (cohort B); their disease must have progressed despite treatment with iodine-131 therapy or they are not candidates for iodine-131 therapy and their disease cannot be completely removed by surgery; all patients with WDTC are expected to be on thyroxine suppression therapy
* Patients must have radiographically or biochemically measurable disease; radiographically measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as >= 20 mm with conventional techniques or as >= 10 mm with spiral computed tomography (CT) scan; biochemically measurable disease is defined as an elevated thyroglobulin (WDTC patients) or calcitonin (MTC patients)
* Patients must have evidence of disease progression (objective growth of existing tumors or rising thyroglobulin or calcitonin levels) within the last 6 months
* Patients cannot have received prior receptor tyrosine kinase inhibitors; patients cannot have received more than one prior chemotherapy regimen for metastatic disease; patients cannot have received prior external beam radiation to the measured tumor constituting the target lesion(s)
* Life expectancy of greater than 12 weeks
* Eastern Cooperative Oncology Group (ECOG) performance status 0-2 (Karnofsky >= 60%)
* Leukocytes >= 3,000/mcL
* Absolute neutrophil count >= 1,500/mcL
* Platelets >= 100,000/mcL
* Hemoglobin >= 9 g/dL
* Serum calcium =< 12.0 mg/dL
* Total serum bilirubin within normal institutional limits
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =< 2.5 X institutional upper limit of normal OR =< 5 X institutional upper limit of normal if patient has liver metastases
* Creatinine within normal institutional limits OR creatinine clearance >= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal
* Patients must have corrected QT interval (QTc) < 500 msec
* The following groups of patients are eligible provided they have New York Heart Association class II (NYHA) cardiac function on baseline echocardiogram (ECHO)/multigated acquisition scan (MUGA):

* Those with a history of class II heart failure who are asymptomatic on treatment
* Those with prior anthracycline exposure
* Those who have received central thoracic radiation that included the heart in the radiotherapy port
* The effects of sunitinib on the developing human fetus at the recommended therapeutic dose are unknown; for this reason and because antiangiogenic agents are known to be teratogenic, women of childbearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; all women of childbearing potential must have a negative pregnancy test prior to receiving sunitinib; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier; at least 4 weeks must have elapsed since any major surgery
* Patients may not be receiving any other investigational agents
* Patients who have received prior treatment with any other antiangiogenic agent (e.g., bevacizumab, sorafenib, pazopanib, AZD2171, PTK787, vascular endothelial growth factor \[VEGF\] Trap, etc.)
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to sunitinib
* Patients with QTc prolongation (defined as a QTc interval equal to or greater than 500 msec), serious ventricular arrhythmia (ventricular fibrillation or ventricular tachycardia greater than or equal to 3 beats in a row) or other significant electrocardiogram (ECG) abnormalities are excluded
* Patients with poorly controlled hypertension (systolic blood pressure of 140 mmHg or higher or diastolic blood pressure of 90 mmHg or higher) are ineligible
* Patients who require use of therapeutic doses of coumarin-derivative anticoagulants such as warfarin are excluded, although doses of up to 2 mg daily are permitted for prophylaxis of thrombosis; Note: Low molecular weight heparin is permitted provided the patient's prothrombin time (PT) international normalized ratio (INR) is =< 1.5
* Patients with any condition (e.g., gastrointestinal tract disease resulting in an inability to take oral medication or a requirement for intravenous \[IV\] alimentation, prior surgical procedures affecting absorption, or active peptic ulcer disease) that impairs their ability to swallow and retain sunitinib tablets are excluded
* Patients with any of the following conditions are excluded:

* Serious or non-healing wound, ulcer, or bone fracture
* History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 28 days of treatment
* Any history of cerebrovascular accident (CVA) or transient ischemic attack within 12 months prior to study entry
* History of myocardial infarction, cardiac arrhythmia, stable/unstable angina, symptomatic congestive heart failure, or coronary/peripheral artery bypass graft or stenting within 12 months prior to study entry
* History of pulmonary embolism within the past 12 months
* Class III or IV heart failure as defined by the NYHA functional classification system
* Because sunitinib is metabolized primarily by the CYP3A4 liver enzyme, the eligibility of patients taking medications that are potent inducers or inhibitors of that enzyme will be determined following a review of their case by the principal investigator; every effort should be made to switch patients taking such agents or substances to other medications, particularly patients with gliomas or brain metastases who are taking enzyme-inducing anticonvulsant agents
* Patients with known brain metastases should be excluded because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events; N.B.: Patients with brain metastases with stable neurologic status following local therapy (surgery or radiation) for at least 8 weeks from definitive therapy and without neurologic dysfunction that would confound the evaluation of neurologic and other adverse events are eligible for participation; patients cannot be receiving enzyme inducing anti-convulsants including carbamazepine, phenobarbital, and phenytoin
* Patients with uncontrolled intercurrent illness including, but not limited to, ongoing or active infections or psychiatric illness/social situations that would limit compliance with study requirements are ineligible
* Pregnant women are excluded from this study because sunitinib is an antiangiogenic agent with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with sunitinib, breastfeeding should be discontinued if the mother is treated with sunitinib malate
* Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with sunitinib; in addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated
* Patients with conditions classified as NYHA III or IV per the New York Heart Association classifications