Pemetrexed and Avelumab in Treating Patients With MTAP-Deficient Metastatic Urothelial Cancer

Inclusion Criteria: * Patients must have histologic confirmation of MTAP-deficient metastatic urothelial carcinoma. MTAP-deficiency must be verified by institutional Clinical Laboratory Improvement Act (CLIA)-certified immunohistochemistry (IHC). Histological variants such as glandular, squamous, sarcomatoid, micropapillary, plasmacytoid, and small cell changes will be allowed for this trial if these tumors are MTAP-deficient. * Patients can be considered for second line of therapy (after chemotherapy or immune checkpoint inhibitor with PD-\[L\]1 agent) or for third line of therapy (can have previously received chemotherapy and immune checkpoint inhibitor with PD-\[L\]1 blockade). Any prior intravesical therapy is allowed and does not count as a prior line of therapy. * Patients who received methotrexate-containing chemotherapy (e.g. methotrexate/vinblastine/adriamycin/cisplatin \[MVAC\]) as neoadjuvant therapy or first-line systemic therapy at least 12 months prior will be allowed for this trial. * All patients must have measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 and tumors of sufficient sizes for biopsy. In general, liver and lung lesions should be at least 1.0 cm, and patients with lymph node-only disease should have lesions of \>= 1.5 cm in shortest dimension. Patients with disease confined to bone may be eligible if a measurable lytic defect is present. The study principal investigator (PI) is the final arbiter in questions related to measurability. Patients with a three-dimensional mass or pelvic sidewall fixation on bladder examination under anesthesia are considered to have measurable disease. * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status =\< 2. * Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) =\< 2.5 x upper limit of normal (ULN), or =\< 5 ULN if documented liver metastases are present. * Total bilirubin =\< 1.5 x ULN, except subjects with Gilbert's syndrome or liver metastases, who must have a baseline total bilirubin =\< 3.0 mg/dL. * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L. * Hemoglobin \>= 9 g/dL (may have been transfused). * Platelets \>= 100 x 10\^9/L. * Normal serum creatinine, or a creatinine clearance \>= 40 ml/min \[either measured using a 24 hour urine, calculated using Cockcroft-Gault, or estimated using the MDRD method from the National Kidney Disease Education Program (NKDEP) (the method reported by MD Anderson Cancer Center \[MDACC\] laboratories). * Negative serum or urine pregnancy test at screening for women of child-bearing potential. * Females of childbearing potential who are sexually active with a non-sterilized male partner and non-sterilized males must use a highly effective method of contraception for 28 days prior to the first dose of investigational product, and must agree to continue using such precautions for 180 days after the final dose of investigational product; cessation of contraception after this point should be discussed with a responsible physician. They must also refrain from egg cell donation for 180 days after the final dose of investigational product. * Non-sterilized males who are sexually active with a female partner of childbearing potential must use a highly effective method of contraception from days 1 through 180 post last dose. In addition, they must refrain from sperm donation for 180 days after the final dose of investigational product. * The ability to interrupt nonsteroidal anti-inflammatory drugs (NSAIDS) 2 days before (5 days for long-acting NSAIDs), the day of, and 2 days following administration of pemetrexed. * The ability to take folic acid, vitamin B12, and dexamethasone according to protocol. * Mild autoimmune conditions (such as localized psoriasis) requiring minimal treatment or systemic autoimmune conditions well controlled by target agents such as an anti-IL-17 that do not affect overall immune system. Patients with a history of Hashimoto's thyroiditis only requiring hormone replacement, type I diabetes, or conditions not expected to recur in the absence of an external trigger are allowed to participate. Exclusion Criteria: * Primary central nervous system (CNS) malignancies or CNS metastases, including leptomeningeal metastases, are not allowed. Subjects with previously treated brain metastases will be allowed if the brain metastases have been stable for at least 4 weeks following prior treatment and no ongoing steroid requirement. * Any other malignancy from which the patient has been disease-free for less than 2 years, except for non-melanomatous skin cancer, controlled localized prostate cancer, in situ carcinoma of any site. * Women who are pregnant or breastfeeding or intend to become pregnant during their participation in the study. * Presence of third space fluid which cannot be controlled by drainage. For patients who develop or have baseline clinically significant pleural or peritoneal effusions (on the basis of symptoms or clinical examination) before or during initiation of pemetrexed therapy, consideration should be given to draining the effusion prior to dosing. However, if, in the investigator's opinion, the effusion represents progression of disease, the patient should be discontinued from study therapy. * Known or suspected autoimmune disease. Patients with a history of inflammatory bowel disease (including Crohn's disease and ulcerative colitis) and autoimmune disorders such as rheumatoid arthritis, systemic progressive sclerosis (scleroderma), systemic lupus erythematosus or autoimmune vasculitis (e.g., Wegener's granulomatosis) are excluded from this study. * Any condition requiring systemic treatment with corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days prior to first dose of study drug. Inhaled steroids and adrenal replacement steroids doses \> 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease. * History of primary immunodeficiency. * Patients who have prior organ transplantation, including allogeneic stem-cell transplant. * Vaccinations within 4 weeks of the first dose of avelumab and while on trials is prohibited except for administration of inactivated vaccines. * True positive test results for hepatitis A, B, or C during screening. * Known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS). * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, current pneumonitis, symptomatic congestive heart failure, unstable angina pectoris, symptomatic cardiac arrhythmia, or interstitial lung disease. * 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. * Persisting toxicity related to prior therapy (National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v.\] 5.0 grade \> 1); however, alopecia, sensory neuropathy grade =\< 2, or other grade =\< 2 not constituting a safety risk based on investigator's judgment are acceptable. * Uncontrolled psychiatric illness/social situations that would limit compliance with study requirements or compromise the ability of the subject to give written informed consent. * Known allergy or hypersensitivity to study drug formulations. * Major surgical procedure (as defined by the PI or co-PIs within 28 days prior to the first dose of therapy) or still recovering from prior surgery. * Patient currently on dialysis.