A Pilot Study to Investigate the Safety and Clinical Activity of Avelumab (MSB0010718C) in Thymoma and Thymic Carcinoma After Progression on Platinum-Based Chemotherapy

* INCLUSION CRITERIA: * Participants must have histologically confirmed thymoma or thymic carcinoma by the pathology department/CCR/NCI. * Participants must have had at least one prior line of platinum-based chemotherapy or participant must have refused cytotoxic chemotherapy. Progressive disease must be documented prior to study entry and participants must have advanced, unresectable disease that is not amenable to surgical resection. * Prior treatment of PD-1 or PD-L1-directed immune checkpoint blockade is permitted if treatment was not discontinued due to disease progression or lifethreatening adverse events per the investigators discretion (laboratory abnormalities alone with prior therapy will not exclude participants from this trial). * Disease must be measurable with at least 1 unidimensional measurable lesion by RECIST 1.1. * Individuals aged greater than or equal to 18 years \-- Because no dosing or adverse event data are currently available on the use of Avelumab in participants \<18 years of age, children are excluded from this study, but will be eligible for future pediatric trials. * ECOG performance status less than or equal to 1. * Participants must have normal organ and marrow function as defined below: * absolute neutrophil count: greater than or equal to 1,500/mm\^3 OR greater than or equal to 1.5 x 10\^9/L * platelets greater than or equal to 100,000/mm\^3 OR greater than or equal to 100 x 10\^9/L * hemoglobin greater than or equal to 9g/dL (may have been transfused) * total bilirubin less than or equal to 1.5 x the upper limit of normal range(ULN) * AST(SGOT)/ALT(SGPT) less than or equal to 2.5 x ULN OR less than or equal to 5 x ULN for participants with documented metastatic disease to the liver * creatinine clearance greater than or equal to 30 mL/min according to the Cockcroft Gault formula (or local institutional standard method) * Highly effective contraception for all individuals if the risk of conception exists. (Note: The effects of the study drug on the developing human fetus are unknown. Thus, individuals of childbearing potential and those able to father a child must agree to use highly effective contraception, defined as 2 barrier methods, or 1 barrier method with a spermicide, an intrauterine device or use of oral female contraceptive. Effective contraception must be used 30 days prior to first study drug administration, for the duration of trial participation, and at least 30 days after last avelumab treatment administration if the risk of conception exists. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this trial, the treating physician should be informed immediately.) * Pregnancy test: negative serum or urine pregnancy test at screening for women of childbearing potential. * Ability of individual to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: * Concurrent treatment with a non-permitted drug * Concurrent anticancer treatment within 28 days before the start of trial treatment (e.g., cytoreductive therapy, radiotherapy \[with the exception of palliative bone directed radiotherapy\], immune therapy, or cytokine therapy except for erythropoietin); major surgery within 28 days before the start of trial treatment (excluding prior diagnostic biopsy); concurrent systemic therapy with immunosuppressive agents within 28 days before the start of trial treatment; use of hormonal agents for the treatment of thymic cancer within 7 days before the start of trial treatment; or use of any investigational drug within 28 days before the start of trial treatment. --Note: Individuals receiving bisphosphonate or denosumab are eligible provided treatment was initiated at least 14 days before the first dose of avelumab. * History of previous malignant disease within the last 2 years with the following exceptions: basal or squamous cell carcinoma of the skin, cervical carcinoma in situ, ductal carcinoma in situ of the breast, papillary or follicular thyroid carcinoma, and non-muscle invasive bladder cancer. * Active autoimmune disease that might deteriorate when receiving an immune-stimulatory agent. Participants with diabetes type 1, vitiligo, psoriasis, pure red cell aplasia, Good s syndrome or hypo- or hyperthyroid diseases not requiring immunosuppressive treatment are eligible. Anti-acetylcholine receptor antibodies will be checked during screening. Participants will be ineligible if results are positive, even if there is no clinical history of autoimmune disease. * Participants with symptomatic brain metastases will be excluded from trial secondary to poor prognosis. However, participants who have had treatment for their brain metastasis and whose brain disease has remained stable for 3 months without steroid therapy may be enrolled. * Active infection requiring systemic therapy or significant acute or chronic infections including, among others: * Hepatitis B virus (HBV) or hepatitis C virus (HCV) infection at screening (positive HBV surface antigen or HCV RNA if anti-HCV antibody screening test positive). * Known history of testing positive for HIV or known acquired immunodeficiency syndrome. * Prior organ transplantation including allogenic stem-cell transplantation. * Known prior severe hypersensitivity to investigational product or any component in its formulations, including known severe hypersensitivity reactions to monoclonal antibodies (NCI CTCAE v5 Grade greater than or equal to 3). * Persisting toxicity related to prior therapy (NCI CTCAE v5Grade \> 1) however, alopecia, sensory neuropathy Grade less than or equal to 2, or other Grade less than or equal to 2 not constituting a safety risk based on investigator s judgment are acceptable. * Pregnancy or lactation period. Note: a negative pregnancy test is required for individuals of childbearing potential. Individuals who are postmenopausal (age-related amenorrhea greater than or equal to 12 consecutive months or follicle-stimulating hormone (FSH) \> 40 milli international units per milliliter \[mIU/ml\]), or who had undergone hysterectomy or bilateral oophorectomy are exempt from pregnancy testing. If necessary to confirm postmenopausal status a FSH level will be included at screening. * Pregnant individuals are excluded from this study because Avelumab is in the class of agents known as antineoplastics/monoclonal antibodies with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with Avelumab, breastfeeding should be discontinued if the mother is treated with Avelumab. * Known alcohol or drug abuse. * Clinically significant (i.e. active) cardiovascular disease: cerebral vascular accident/stroke (\< 6 months prior to enrollment), myocardial infarction (\< 6 months prior to enrollment), unstable angina, congestive heart failure (\>= New York Heart Association Classification Class II), or serious cardiac arrhythmia requiring medication. * All other severe acute or chronic medical conditions including immune colitis, inflammatory bowel disease, immune pneumonitis, pulmonary fibrosis or psychiatric conditions including recent (within the past year) or active suicidal ideation or behavior; or laboratory abnormalities that may increase the risk associated with study participation or study treatment administration or may interfere with the interpretation of study results and, in the judgment of the investigator, would make the participants inappropriate for entry into this study. * Any psychiatric condition that would prohibit the understanding or rendering of informed consent. * Legal incapacity or limited legal capacity. * Non-oncology vaccine therapies for prevention of infectious disease (e.g. seasonal flu vaccine, human papilloma virus vaccine) within 4 weeks of study drug administration with the exception of vaccinations against COVID-19. Vaccination while on study is also prohibited except for administration of inactivated vaccines (e.g. inactivated influenza vaccines) and vaccinations against COVID-19. * HIV-positive TET participants are ineligible because of the risk of developing opportunistic infections after treatment with an immune checkpoint inhibitor. Prior cases of disseminated herpes virus and fungal infections have been documented in this patient population.