Lenvatinib and Pembrolizumab to Treat Patients With Anal or Rectum Cancer That Has Gotten Worse After Initial Treatment

Inclusion Criteria: * Written informed consent and HIPAA authorization for release of personal health information prior to registration. * Age ≥ 18 years at the time of consent. * The Eastern Cooperative Oncology Group Performance Status of 0-1 within 7 days prior to registration. * Histological confirmation of anorectal squamous cell carcinoma per the American Joint Committee on Cancer 8th edition. NOTE: If archived tissue is not available for diagnostic histological confirmation \[core, incisional, or excisional\], a new biopsy of a tumor lesion prior to tumor irradiation should be obtained. * Unresectable locally advanced or metastatic anorectal squamous cell carcinoma following progression on first line chemotherapy or chemoradiation therapy. * Measurable disease based on Response Evaluation Criteria In Solid Tumors 1.1 Lesions situated in a previously irradiated area are considered measurable if progression has been demonstrated in such lesions within 28 days prior to registration. * Demonstrate adequate organ function as defined in the table below. All screening labs to be obtained within 28 days prior to registration. * System / Laboratory Value 1. Hematological 1. Absolute Neutrophil Count (ANC): ≥1500/µL 2. Platelets: ≥100 000/µL 3. Hemoglobin (Hgb): ≥9.0 g/dL or ≥5.6 mmol/La 2. Renal 1. Creatinine: ≤1.5 × ULN -OR- 2. Measured or calculated creatinine clearance (GFR can also be used in place of creatinine or CrCl): 5.1.9 ≥30 mL/min for participant with creatinine levels \>1.5 × institutional ULN 3. Hepatic 1. Total Bilirubin: ≤1.5 ×ULN OR direct bilirubin ≤ULN for participants with total bilirubin levels \>1.5 × ULN 2. Aspartate aminotransferase (AST): ≤2.5 × ULN (≤5 × ULN for participants with liver metastases) 3. Alanine aminotransferase (ALT): ≤2.5 × ULN (≤5 × ULN for participants with liver metastases) 4. Coagulation 1. International Normalized Ratio (INR) or Prothrombin Time (PT): ≤1.5 × ULN unless participant is receiving anticoagulant therapy as long as PT or aPTT is within therapeutic range of intended use of anticoagulants 2. Activated Partial Thromboplastin Time (aPTT): ≤1.5 × ULN unless participant is receiving anticoagulant therapy as long as PT or aPTT is within therapeutic range of intended use of anticoagulants (Criteria must be met without erythropoietin dependency and without packed red blood cell \[pRBC\] transfusion within last 2 weeks. Creatinine clearance \[CrCl\] should be calculated per institutional standard.) * Females of childbearing potential must have a negative serum pregnancy test within 7 days prior to registration. * Females of childbearing potential who are sexually active with a male able to father a child must be willing to abstain from heterosexual activity or to use an effective method(s) of contraception. Males able to father a child who are sexually active with female of childbearing potential must be willing to abstain from heterosexual activity or to use an effective method(s) of contraception. * Hepatitis B positive subjects: Participants who are known to be hepatitis B surface antigen positive are eligible if they have received hepatitis B virus antiviral therapy for at least 4 weeks and have undetectable HBV viral load prior to treatment. Participants should remain on anti-viral therapy throughout study intervention and follow local guidelines for hepatitis B virus anti-viral therapy post completion of study intervention. Prospective testing is not required for participants of unknown status unless mandated by local policy. * Hepatitis C positive subjects: Participants with history of hepatitis C virus infection are eligible if hepatitis C virus viral load is undetectable at screening. Participants must have completed curative anti-viral therapy at least 4 weeks prior to study registration. Prospective testing is not required for participants of unknown status unless mandated by local policy. * Human Immunodeficiency Virus-infected participants must have well-controlled human immunodeficiency virus on antiretroviral therapy, defined as: 1. Participants on antiretroviral therapy must have a CD4+ T-cell count ≥350 cells/mm3 at the time of screening. 2. Participants on antiretroviral therapy must have achieved and maintained virologic suppression defined as confirmed human immunodeficiency virus Ribonucleic Acid level below 50 or the LLOQ (below the limit of detection) using the locally available assay at the time of screening and for at least 12 weeks before screening. 3. It is advised that participants must not have had any acquired immunodeficiency syndrome-defining opportunistic infections within the past 12 months. 4. Participants on antiretroviral therapy must have been on a stable regimen, without changes in drugs or dose modification, for at least 4 weeks before study entry (Day 1) and agree to continue antiretroviral therapy throughout the study 5. Prospective testing is not required for participants of unknown status unless mandated by local policy. * Have adequately controlled blood pressure with or without antihypertensive medications, defined as blood pressure ≤150/90 mm Hg with no change in antihypertensive medications within 1 week prior to registration. * Ability of the subject to understand and comply with study procedures for the entire length of the study, as determined by the enrolling physician or protocol designee. Exclusion Criteria: * Active infection requiring systemic therapy. * Has received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent or with an agent directed to another stimulatory or co-inhibitory T-cell receptor (eg, CTLA-4, OX-40, CD137). * Has received prior systemic anti-cancer therapy, including an investigational agent(s) or investigational device, within 4 weeks prior to study registration. * Has received prior radiotherapy within 2 weeks of study registration. Participants must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had radiation pneumonitis. A 1-week washout is permitted for palliative radiation (defined as ≤2 weeks of radiotherapy) to non-CNS disease. * Has received a live vaccine or live-attenuated vaccine within 30 days before study registration. Administration of killed vaccines is allowed. * 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 study registration. * Known additional malignancy that is progressing or has required active treatment within the past 3 years. Note: Participants with basal cell carcinoma of the skin, squamous cell carcinoma of the skin or carcinoma in situ (e.g., breast carcinoma in situ, cervical cancer in situ), excluding carcinoma in situ of the bladder, that have undergone potentially curative therapy are not excluded. * Has known active central nervous system metastases and/or carcinomatous meningitis. Participants with previously treated brain metastases may participate provided they are radiologically stable, i.e. without evidence of progression for at least 4 weeks by repeat imaging (note that the repeat imaging should be performed during study screening), clinically stable and without requirement of steroid treatment for at least 14 days prior to study registration. * Has severe hypersensitivity (≥Grade 3) to pembrolizumab, lenvatinib and/or any of its excipients. * 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 and is allowed. * Has a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease. * Has a history or current evidence of any condition, therapy, or laboratory abnormality or other circumstance that might confound the results of the study, interfere with the participant's participation for the full duration of the study, such that it is not in the best interest of the participant 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 breastfeeding or expecting to conceive or father children within the projected duration of the study, starting with the screening visit through 120 days after the last dose of trial treatment. * Has had an allogeneic stem cell/solid organ transplant. * Has had major surgery within 3 weeks prior to study registration. Note: Adequate wound healing after major surgery must be assessed clinically, independent of time elapsed for eligibility. * Has preexisting ≥Grade 3 gastrointestinal or non-gastrointestinal fistula. * Has urine protein ≥1 g/24 hours. Note: Participants with proteinuria ≥2+ (≥100 mg/dL) on urine dipstick testing (urinalysis) will undergo 24-hour urine collection for quantitative assessment of proteinuria. * Has a left ventricular systolic function below the institutional (or local laboratory) normal range, as determined by multigated acquisition (MUGA) or echocardiogram (ECHO). * Has radiographic evidence of encasement or invasion of a major blood vessel, or of intratumoral cavitation. Note: the degree of proximity to major blood vessels should be considered because of the potential risk of severe hemorrhage associated with tumor shrinkage/necrosis following lenvatinib therapy. * Prolongation of QTcF interval to \>480 ms. If the QTcF is prolonged to \>480 ms in the presence of a pacemaker, contact the Lead Principal Investigator to determine eligibility. * Has clinically significant cardiovascular disease within 12 months from study registration, including New York Heart Association Class III or IV congestive heart failure, unstable angina, myocardial infarction, cerebral vascular accident, or cardiac arrhythmia associated with hemodynamic instability. Note: Medically controlled arrhythmia would be permitted. * Gastrointestinal malabsorption or any other condition that might affect the absorption of lenvatinib. * Active hemoptysis (bright red blood of at least 0.5 teaspoon) within 3 weeks prior to study registration.