Avelumab in Combination With Fluorouracil and Mitomycin or Cisplatin and Radiation Therapy in Treating Participants With Muscle-Invasive Bladder Cancer

Inclusion Criteria: * Histologic proof of T2-T4a N0M0 (American Joint Committee on Cancer \[AJCC\] 8th edition) with predominant urothelial carcinoma. Mixed histologies are acceptable provided urothelial carcinoma is the predominant histology. Small cell urothelial carcinoma is excluded. * Cystoscopy with maximal TURBT performed =\< 70 days of study registration. NOTE: Both completely resectable or partially resectable tumors are eligible as long as the treating urologist attempted complete resection. Exam under anesthesia needs to be performed and documented. * Absolute neutrophil count (ANC) \>= 1500/mm\^3 =\< 28 days prior to registration. * Platelets (PLT) 100,000/mm\^3 =\< 28 days prior to registration. * Total bilirubin =\< 1.5 upper limit of normal (ULN) =\< 28 days prior to registration. * Aspartate transaminase (AST) =\< 2.5 x ULN (=\< 5 x ULN for patients with liver involvement) =\< 28 days prior to registration. * Alanine aminotransferase (ALT) =\< 2.5 x ULN (=\< 5 x ULN for patients with liver involvement) =\< 28 days prior to registration. * Hemoglobin (Hgb) \>= 9 gm/dl =\< 28 days prior to registration. * Calculated creatinine clearance must be \>= 30 ml/min using the Cockcroft-Gault formula =\< 28 days prior to registration. * Eastern Cooperative Oncology Group (ECOG) performance status (PS 0, 1, 2). * Ability to provide informed written consent. * Willing to return to enrolling institution for follow-up (during the active monitoring phase of the study). * Life expectancy \>= 6 months. * Negative serum pregnancy test done =\< 14 days prior to registration, for women of childbearing potential only. Exclusion Criteria: * Patients with locally advanced unresectable (T4b) or metastatic urothelial carcinoma (N1M0-1) as assessed on baseline radiographic imaging obtained =\< 70 days prior to study registration. The required radiographic imaging includes: * Abdomen/pelvis computed tomography (CT) or magnetic resonance imaging (MRI) scan * Chest x-ray or CT scan. * Patients with concurrent urothelial carcinoma and/or related variants anywhere outside bladder * NOTE: Patients with history of non-invasive (Ta, Tis) upper tract urothelial carcinoma that has been definitively treated with at least one post-treatment disease assessment (i.e. cytology, biopsy, imaging) that demonstrates no evidence of residual disease are eligible. * A prior or concurrent malignancy of any other site or histology unless the patient has been disease-free for \> 2 years prior to registration except for: * Non-melanoma skin cancer and/or localized prostate cancer (T2 a or b , Gleason \< 3+4) or carcinoma in situ of the uterine cervix which has been adequately treated =\< 2 years prior to registration * Or undergoing active surveillance per standard-of-care management (e.g., chronic lymphocytic leukemia Rai stage 0, prostate cancer with Gleason score =\< 3+4, and prostate-specific antigen \[PSA\] =\< 10 mg/mL, etc.). * Patients who have received the last administration of an anti-cancer therapy including chemotherapy, immunotherapy, and monoclonal antibodies =\< 4 weeks prior to registration, or who have not recovered from the side effects of such therapy. * EXCEPTION: Except single dose intravesical chemotherapy administered after TURBT. * Patients who have received prior therapy with immune checkpoint inhibitors (e.g. anti-PD-1, anti-PD-L1, anti-LAG3, anti-CTLA-4, anti-TIM3) or immune co-stimulatory molecules (e.g. anti-CD137, anti-OX40, anti-GITR) directed agents. * Patients who have undergone major surgery (e.g. intra-thoracic, intra-abdominal or intra-pelvic), open biopsy or significant traumatic injury =\< 4 weeks prior to registration, or who have not recovered from side effects of such procedure or injury prior to registration. * NOTE: Patients who have had minor procedures (i.e. TURBT) or percutaneous biopsies prior to registration are eligible. * Patients with history of cirrhosis, alcoholic or non-alcoholic steatohepatitis (NASH), auto-immune hepatitis, or previous grade 3-4 drug-related hepatitis. * Patient with history of prior solid organ or allogeneic bone marrow transplant. * Clinically significant cardiac diseases, including any of the following: * History or presence of serious uncontrolled ventricular arrhythmias. * Clinically significant resting bradycardia. * Any of the following =\< 3 months prior to registration: myocardial infarction (MI), severe/unstable angina, coronary artery bypass graft (CABG), congestive heart failure (CHF), cerebrovascular accident (CVA), transient ischemic attack (TIA), pulmonary embolism (PE). * Uncontrolled hypertension defined by a systolic blood pressure (SBP) \>= 160 mm Hg and/or diastolic blood pressure (DBP) \>= 100 mm Hg, with or without anti-hypertensive medication(s). * History of untreated human immunodeficiency virus (HIV) * NOTE: There is no requirement to screen patients for HIV. Patients with history of HIV infection are allowed if on effective highly active antiretroviral therapy (HAART) therapy and CD4 count more than 250. * History of active hepatitis B infection * NOTE: There is no requirement to screen patients for hepatitis B. * Known diagnosis of any condition (i.e. post-hematopoietic or organ transplant, rheumatoid arthritis, systemic lupus erythematosus, inflammatory bowel disease, etc.) that requires chronic immunosuppressive therapy. * NOTE: Usage of non-steroidal anti-inflammatory medications (NSAIDS) for the treatment of osteoarthritis and uric acid synthesis inhibitors for the treatment of gout are permitted. For questions, please consult the study chair. * Other concurrent severe and/or uncontrolled concomitant medical conditions (e.g. active or uncontrolled infection, uncontrolled diabetes) that could cause unacceptable safety risks or compromise compliance with the protocol. * Pregnant or breast-feeding women. * Women of child-bearing potential, who are biologically able to conceive, and not employing two forms of highly effective contraception. Highly effective contraception must be used throughout the trial and up to 8 weeks after the last dose of study drug (e.g. male condom with spermicidal; diaphragm with spermicide; intra-uterine device). Women of child-bearing potential, defined as sexually mature women who have not undergone a hysterectomy or who have not been naturally postmenopausal for at least 12 consecutive months (i.e., who has had menses any time in the preceding 12 consecutive months), must have a negative serum pregnancy test =\< 14 days prior to registration. * Fertile males not willing to use contraception, as stated above. * Patients unwilling or unable to comply with the protocol. * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm. * Known prior severe hypersensitivity to investigational product or any component in its formulations, including known severe hypersensitivity reactions to monoclonal antibodies (National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v\] 5.0 grade \>= 3).