Lenvatinib and Pembrolizumab in People With Advanced Soft Tissue Sarcoma

Inclusion Criteria: * Male or female age ≥18 years at time of informed consent * Be willing and able to provide written informed consent/assent for the trial * Be willing to comply with treatment protocol * Availability of archival tissue for correlative studies; either a paraffin block or at least 20 unstained slides are acceptable * Patients must have a metastatic and/or unresectable soft tissue sarcoma as below: * Cohort A: Leiomyosarcoma * Cohort B: High grade undifferentiated pleomorphic sarcoma * Cohort C: Vascular sarcomas (including angiosarcoma and epithelioid hemangioendothelioma) * Cohort D: Other soft tissue sarcomas (including synovial sarcoma and malignant peripheral nerve sheath tumor * Cohort E: Bone sarcomas (including osteosarcoma and chondrosarcoma) * Subjects must have had at least 1 but not more than 3 prior lines of systemic therapy (e.g. chemotherapy, immunotherapy, targeted or biological therapy); patients who decline the standard of care first-line systemic therapy will be eligible for this trial. Prior adjuvant therapy will not count provided it was completed more than 6-month previously. * Presence of measurable disease per RECIST v1.1. Target lesions must not be chosen from a previously irradiated field unless there has been radiographically and/or pathologically documented tumor progression in that lesion prior to enrollment.Where and when possible, target lesions will not be chosen as the biopsy lesion. * Must have a performance status ECOG 0-1. * Screening laboratory values must meet the following criteria: * Neutrophils ≥ 1000/μL * Platelets ≥ 100x10\^3/μL * Hemoglobin \> 9.0 g/dL (without packed red blood cell \[pRBC\] transfusion within the last 2 weeks) * AST and ALT ≤ 2.5 x ULN (if liver metastases are present, AST and ALT≤ 5x ULN) * Total Bilirubin ≤ 1.5 x ULN OR direct bilirubin \< ULN for participants with total bilirubin levels \>1.5 x ULN (except participants with Gilbert syndrome, who can have a total bilirubin \< 3.0 mg/dL) * Serum creatinine ≤ 1.5 x ULN OR creatinine clearance (CrCl) ≥ 40 mL/min per the Cockcroft-Gault formula if creatinine is \>1.5 x ULN * INR OR prothrombin time (PT) \< 1.5, unless participant is receiving anticoagulant therapy as long as PT or aPTT is within therapeutic range of intended use of anticoagulants * Male participants must agree to use adequate contraception (complete abstinence, male condom) of this protocol during the treatment period and for at least 120 days after the last dose of study therapy and refrain from donating sperm during this period. * A female participant is eligible to participate if she is not pregnant (for women of childbearing potential, serum pregnancy test must be negative within 72 hours prior to initiation; not breast feeding, and at least one of the following conditions applies: * Not a woman of child bearing potential including: * Premenopausal with one of the following: documented hysterectomy, documented bilateral salpingectomy, documented bilateral oophorectomy * Postmenopausal females defined as no menses for 12 months without an alternative medical cause (a high follicle stimulating hormone level in the postmenopausal range may be used to confirm a post-menopausal state in women not using hormonal contraception or hormonal replacement therapy (HRT). However, in the absence of 12 months of amenorrhea, confirmation with two FSH measurements in the postmenopausal range is required. * A woman of child bearing potential who agrees to highly effective contraception from the start of therapy through 120 days after the last dose of study medication * Participants must be able to swallow and retain oral medication or have a functioning G-tube in place. Exclusion Criteria: * Untreated metastatic brain (subjects with treated brain metastases will be eligible, provided that they are radiographically stable, i.e. without evidence of progression for at least 4 weeks by repeat imaging performed during study screening, clinically stable and without requirement of steroid treatment for at least 14 days prior to the first dose of study treatment). * Concurrent anti-cancer therapy (chemotherapy, definitive radiation therapy, surgery, immunotherapy, biologic therapy or tumor embolization) other than study treatment. Concurrent therapy with bisphosphonates or denosumab for bone metastases is allowed, provided they are started prior to study entry. Palliative radiation to non-target lesions is also allowed as screened by the Principal Investigator. * Presence of any other concurrent malignancy requiring active therapy or thought to potentially interfere with the safe conduct or assessment of outcomes on this trial. * History of allergy or intolerance to lenvatinib or study drug components (or any of their excipients) or severe (\> Grade 3) hypersensitivity reaction to pembrolizumab and/or any of its excipients or any monoclonal antibody. * Prior use of lenvatinib or pazopanib or any PD-1/PD-L1 or anti-PD-L2 targeted therapies or with an agent directed at another stimulatory or co-inhibitory T-cell receptor (CTLA-4, OX-40, CD137). * Uncontrolled hypertension (systolic pressure \>140mm Hg or diastolic pressure \>90mm Hg, despite optimal medical management. * Prior systemic anti-cancer therapy including use of another investigation drug or device (i.e., outside study treatment) during, or within 3 weeks of trial entry (time of initiation of experimental drug). * Prior radiotherapy within 2 weeks of the start of study treatment. Participants must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had radiation pneumonitis. * Participants must have recovered all AEs due to previous therapies to ≤ Grade 1 or baseline. Participants with ≤ Grade 2 neuropathy or alopecia may be eligible. If participant received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting study treatment. * Clinically significant proteinuria: °Subjects having \>1+ proteinuria on urinalysis will undergo 24-hour urine collection for quantitative assessment of proteinuria. Subjects with proteinuria ≥/24-hour will be ineligible. * Gastrointestinal malabsorption, gastrointestinal anastomosis, or any other condition that might affect the absorption of lenvatinib. * ≥ Grade 3 gastrointestinal or non-gastrointestinal fistula * New York Heart Association congestive heart failure of grade II or above, unstable angina, myocardial infarction within the past 6 months, serious cardiac arrhythmia associated with significant cardiovascular impairment within the past 6 months (baseline echocardiogram is not required unless clinically indicated) or left ventricular ejection fraction \<55% as determined by echocardiogram. * Subjects with thrombotic, embolic, venous or arterial events, such as cerebrovascular accidents (including transient ischemic attacks), deep venous thrombosis or pulmonary embolism within 6 months of study treatment start. * Therapeutic anticoagulation with Vitamin-K antagonists (e.g., warfarin) is not allowed if the medication dose and/or INR is not considered stable by the treating physician. If the dose and/or INR is stable, therapeutic anticoagulation with Vitamin-K antagonists is allowed with close monitoring. Anticoagulation with heparin or low molecular weight heparin is allowed. * Prolongation of QTc \>480 msec * Any hemorrhage or bleeding event ≥ NCI CTCAE v5.0 Grade ≥3 within 4 weeks prior to start of study medication. * Active infection (any infection requiring systemic treatment) * Subject is known to be positive for Human Immunodeficiency Virus (HIV) or active Hepatitis C Virus (HCV) or active hepatitis B (HBV) infection (positive viral load). Testing for HIV, HCV, or HBV prior to initiation of the study drug is not required. If patient's have a known history of treated HCV, then a viral load is required to confirm clearance of infection. * Serious non-healing wound, ulcer or bone fracture, that is not tumor related. * History of organ allograft (including corneal transplant). * Has a history or current evidence of any medical or other condition, therapy or laboratory abnormality which, in the opinion of the investigator, might confound the results of the study, or preclude participation in a clinical study. * 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 study treatment. The use of physiologic doses of corticosteroids (up to 10 mg/d of prednisone or equivalent) may be approved after consultation with the Primary Investigator. * Active autoimmune disease that has required systemic treatment in the past 2 years (i.e., with the use of disease modifying agents, corticosteroids or immunosuppressive drugs), with the exception of autoimmune thyroid disease, vitiligo, type 1 diabetes mellitus, or psoriasis. 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 a history of (non-infectious) pneumonitis that required steroids or current pneumonitis. * Renal failure requiring active hemo- or peritoneal dialysis. * Has received a live-virus vaccination within 30 days of planned treatment start. Examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, varicella/zoster (chicken pox), yellow fever, rabies, Bacillus Calmette-Guérin (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 known psychiatric or substance abuse disorders that would interfere with the cooperation with the requirements of the trial.