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

This phase II trial studies how well lenvatinib and pembrolizumab work in treating patients with anaplastic thyroid cancer that is stage IVB and has spread to nearby tissue or lymph nodes (locally advanced) and cannot be removed by surgery (unresectable), or stage IVC that has spread to other places in the body (metastatic). Lenvatinib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Immunotherapy with monoclonal antibodies, such as pembrolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Given lenvatinib and pembrolizumab may work better than giving either one alone in treating stage IVB or C 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 \