Pembrolizumab in Recurrent or Metastatic Medullary Thyroid Cancer

* INCLUSION CRITERIA: * Diagnosis: Patients must have histologically confirmed medullary thyroid cancer by the Laboratory of Pathology or a pathology report and history consistent with medullary thyroid cancer. It is not uncommon for a secondary, minor pathologic focus of another form of thyroid cancer to be coincidentally found in 15-20% of patients with medullary thyroid cancer. In such cases, eligibility is based on the discretion of the investigator. * Patients must have evidence of metastatic medullary thyroid cancer including disease that is evaluable on bone, computed tomography (CT) scan or magnetic resonance imaging (MRI). (Patients who are surgical candidates and potentially rendered disease free with surgical resection are not eligible.) * Patients must have elevated calcitonin levels greater than 40 pg/mL * Patients must have minimal or no disease related-symptoms (Minimal symptoms will include those that do not affect activities of daily living or pain that does not require regularly scheduled narcotics.) * Patients must have evaluable disease on imaging * No history of seizures, encephalitis, or multiple sclerosis. * Age greater than or equal to 18 years * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 at study entry (Karnofsky greater than or equal to 70). * Female subject of childbearing potential should have a negative urine or serum pregnancy within 72 hours prior to receiving the first dose of study medication. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. * Female subjects of childbearing potential must be willing to use an adequate method of contraception, Contraception, for the course of the study through 120 days after the last dose of study medication. Note: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the subject * Male subjects of childbearing potential must agree to use an adequate method of contraception. Contraception, starting with the first dose of study therapy through 120 days after the last dose of study therapy. Note: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the subject * Willing to travel to the National Institutes of Health (NIH) for follow-up visits * Able to understand and sign informed consent. * Demonstrate adequate organ function, all screening labs should be performed within 10 days of treatment initiation. * Adequate Organ Function Laboratory Values * Hematological ---Absolute neutrophil count (ANC) greater than or equal to1,000 /mcL * Platelets greater than or equal to 100,000 / mcL * Hemoglobin greater than or equal to 9 g/dL or greater than or equal to 5.6 mmol/L without transfusion or erythropoietin (EPO) dependency (within 7 days of assessment) * Renal * Serum creatinine less than or equal to1.5 X upper limit of normal (ULN) OR * Measured or calculated\* creatinine clearance (Glomerular filtration rate (GFR) can also be used in place of creatinine or creatinine clearance (CrCl) greater than or equal to 60 mL/min for subject with creatinine levels \> 1.5 X institutional ULN * Hepatic * Serum total bilirubin less than or equal to 1.5 X ULN OR Direct bilirubin less than or equal to ULN for subjects with total bilirubin levels \> 1.5 ULN * Aspartate aminotransferase (AST) Serum glutamic-oxaloacetic transaminase (SGOT) and Alanine aminotransferase (ALT) Serum glutamic pyruvic transaminase (SGPT) less than or equal to 2.5 X ULN OR less than or equal to 5 X ULN for subjects with liver metastases * Albumin \>2.5 mg/dL * Coagulation * International Normalized Ratio (INR) or Prothrombin Time (PT) less than or equal to1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants * Activated Partial Thromboplastin Time (aPTT) less than or equal to1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants * Creatinine clearance should be calculated per Cockcroft-Gault equation EXCLUSION CRITERIA: * Is currently participating and receiving study therapy or has participated in a study of an investigational agent and received study therapy or used an investigational device within 4 weeks of the first dose of treatment. * Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. * Has a known history of active TB (Bacillus Tuberculosis) * Hypersensitivity to pembrolizumab or any of its excipients. * Has had a prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study Day 1 or who has not recovered (i.e., less than or equal to Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier. * Has had prior targeted small molecule therapy, or radiation therapy within 2 weeks prior to study Day 1 or who has not recovered (i.e., less than or equal to Grade 1 or at baseline) from adverse events due to a previously administered agent. Note: Subjects with less than or equal to Grade 2 neuropathy are an exception to this criterion and may qualify for the study. Note: If subject received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. * Has active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. * Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Subjects with previously treated brain metastases may participate provided they are stable for 6 months (without evidence of progression by imaging for at least four weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to trial treatment. This exception does not include carcinomatous meningitis which is excluded regardless of clinical stability. * Has history of (non-infectious) pneumonitis that required steroids, evidence of interstitial lung disease or active, non-infectious pneumonitis. * Has an active infection requiring systemic therapy. * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subjects participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator. * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. * Is pregnant or breast feeding or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment. * Has received prior therapy with an anti-Programmed cell death protein 1 (PD-1), anti- programmed cell death-1 ligand 1 (PD-L1), or anti-Programmed death ligand 2 (PD-L2) agent. * Has Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies). * Has active Hepatitis B (e.g., HBsAg reactive) or Hepatitis C (e.g., hepatitis C virus (HCV) ribonucleic acid (RNA) \[qualitative\] is detected). * Has received a live vaccine within 30 days of planned start of study therapy * Concurrent use of systemic steroids, except for physiologic doses of systemic steroid replacement or local (topical, nasal, eye drops or inhaled) steroid use. Limited doses of systemic steroids (e.g., in patients with exacerbations of reactive airway disease or to prevent intravenous (IV) contrast allergic reaction or anaphylaxis in patients who have known contrast allergies) are allowed. * Serious inter-current medical illness which would interfere with the ability of the patient to carry out the treatment program. * Patients with second malignancy within 3 years of enrollment; Patients curatively treated non-melanoma skin cancers or carcinoma in situ of the bladder, are not excluded. Patients with Multiple endocrine neoplasia type 2 (MEN2) and a history of pheochromocytoma will also not be excluded. In addition, patients with prostate cancer who do not require systemic therapy will not be excluded. (A secondary, minor pathologic focus of another form of thyroid cancer may be coincidentally found in 15-20% of patients with medullary thyroid cancer. In such cases, eligibility is based on the discretion of the investigator.) * Patients with previous history of vandetanib or cabozantinib treatment for more than 28 days of treatment (patients have discontinued treatment for 28 days before enrolling).