Lenvatinib and Pembrolizumab for the Treatment of Stage IVB Locally Advanced and Unresectable or Stage IVC Metastatic Anaplastic Thyroid Cancer

Inclusion Criteria: * Pathologic findings supporting the clinical impression of anaplastic thyroid carcinoma. Diagnosis may include consistent with or suggestive of terminology associated with: anaplastic thyroid carcinoma, undifferentiated carcinoma, squamous carcinoma; carcinoma with spindled, giant cell, or epithelial features; poorly differentiated carcinoma with pleomorphism, extensive necrosis with tumor cells present * Patients deemed to have unresectable locoregional disease or metastatic disease. Patients who are unwilling to undergo surgery or external beam radiation are also eligible. Patients with a BRAFV600E mutation who are unable to receive the Food and Drug Administration (FDA) approved drugs, dabrafenib/trametinib, are eligible as long as this is documented * Total bilirubin ≤ 1.5 x upper limit of normal (ULN). Total bilirubin ≤ 3 x ULN for patients with Gilbert's syndrome * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) ≤ 2.5 x ULN, (5 x ULN for patients with concurrent liver metastases) * Serum creatinine ≤ within 1.5 x ULN * Absolute neutrophil count (ANC) ≥ 1.0 x 10\^9/L * Platelets ≥ 100 x 10\^9/L * Hemoglobin ≥9.0 g/dL or ≥5.6 mmol/L * For patients receiving therapeutic anticoagulation: stable anticoagulant regimen and stable institutional normalized ratio (INR) during the 28 days immediately preceding initiation of study treatment * Subjects must be willing to undergo tumor biopsy prior to and after treatment with lenvatinib/pembrolizumab, unless in the opinion of the treating physician, a biopsy is not feasible or safe * Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 1. Evaluation of ECOG is to be performed within 7 days prior to the date of allocation/randomization * Age \>/= 18 years * The participant (or legally acceptable representative if applicable) provides written informed consent for the trial * A male participant must agree to use a contraception in Appendix A during the treatment period and for at least 6 months after the last dose of study treatment * A female participant is eligible to participate if she is not pregnant (see Appendix A), not breastfeeding, and at least one of the following conditions applies: a.) Not a woman of childbearing potential (WOCBP) OR b.) A WOCBP who agrees to follow the contraceptive guidance in Appendix A during the treatment period and for at least 6 months after the last dose of study treatment Exclusion Criteria: * Uncontrolled blood pressure (systolic blood pressure \[BP\] \> 140 mmHg or diastolic BP \> 90 mmHg) in spite of an optimized regimen of antihypertensive medication * In patients with clinically significant electrolyte abnormalities (in the opinion of the treating physician) specifically, low calcium, potassium and magnesium, must have replacement therapy initiated or modified prior to study entry. * Significant cardiovascular impairment: history of congestive heart failure greater than New York Heart Association (NYHA) class II, unstable angina, myocardial infarction or stroke within 6 months of the first dose of study drug, or cardiac arrhythmia requiring medical treatment at screening * Patients with clinically significant hemoptysis or tumor bleeding within two weeks prior to first dose of targeted therapy. Patients with suspected tracheal or esophageal invasion can be included on a case-by-case basis after a discussion with the principal investigator. The degree of tumor invasion/infiltration of major blood vessels (e.g. carotid artery) should be considered because of the potential risk of severe hemorrhage and tracheoesophageal fistula associated with tumor shrinkage/necrosis following lenvatinib therapy * Major surgery within 3 weeks prior to first dose of study interventions. Note: Adequate wound healing after major surgery must be assessed clinically, independent of time elapsed for eligibility * Patients with open wounds or fistulas are excluded * Subjects having \> 2+ proteinuria on urine dipstick testing unless a 24-hour urine collection for quantitative assessment indicates that the urine protein is \<1 g/24 hours * Untreated brain metastases * Prior chemotherapy within \<1 week prior to study Day 1 or patients who have not recovered (i.e., ≤ Grade 2) from adverse events due to a previously administered agent. * Previous anti-angiogenic targeted therapy is excluded * 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 * History of human immunodeficiency virus (HIV) or active hepatitis B (chronic or acute) or hepatitis C infection. Patients with past or resolved hepatitis B infection (defined as having a negative hepatitis B surface antigen \[HBsAg\] test and a positive anti-HBc \[antibody to hepatitis B core antigen\] antibody test) are eligible. However, patients with past or resolved HBV should be monitored for reactivation by a specialist. Patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA) * Has a diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to the first dose of study drug * Has received a live vaccine within 30 days prior to the first dose of study drug. Examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, varicella/zoster (chicken pox), yellow fever, rabies, Bacillus Calmette-Guerin (BCG), and typhoid vaccine. Seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed; however, intranasal influenza vaccines (e.g., FluMist) are live attenuated vaccines and are not allowed * Has severe hypersensitivity (≥Grade 3) to pembrolizumab and/or any of its excipients * Has had an allogenic tissue/solid organ transplant * Has a LVEF below the institutional (or local laboratory) normal range, as determined by multigated acquisition (MUGA) or echocardiogram (ECHO). * Prolongation of QTcF interval to \>480 ms * Gastrointestinal malabsorption or any other condition that might affect the absorption of lenvatinib * Has a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease. * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial * Females who are breastfeeding or pregnant at Screening or Baseline (as documented by a positive beta-human chorionic gonadotropin \[beta-hCG\] (or human chorionic gonadotropin \[hCG\]) test with a minimum sensitivity of 25 IU/L or equivalent units of beta-hCG \[or hCG\]). A women of childbearing potential (WOCBP) who has a positive urine pregnancy test within 72 hours prior to the first infusion will be excluded. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required