Lurbinectedin With or Without Avelumab in Small Cell Carcinoma of the Bladder (LASER)

* INCLUSION CRITERIA: * Histologically or cytologically confirmed metastatic small cell carcinoma of the bladder (SCCB) or other high grade neuroendocrine tumors (HGNETs) of the urinary tract (which includes renal pelvis, ureter, and urethra, and excludes neuroendocrine tumors of the prostate). Mixed histologies, with any component including SCCB or HGNETs, are eligible for inclusion. * Prior treatment as follows: * For Cohort 1: Participants must have received prior ICIs (PD-1 or PD-L1) or be ineligible for treatment with ICIs. * For Cohort 2: Participants must be ICI naive but eligible to receive them. * Participants must have metastatic disease defined as new or progressive lesions. * Participants must have at least one measurable site of disease, per RECIST 1.1. * Participants must have received, be ineligible, or refused prior platinum/etoposide chemotherapy for SCCB or other HGNET of the urinary tract. Platinum ineligibility is defined as a CrCl \<30, or two or more of the following: CrCl \<50-60, ECOG \>=2, hearing loss \>= grade 2, peripheral neuropathy \>= grade 2, New York Heart Association (NYHA) heart failure class \>= class III. * Age \>=18 years. * Eastern Cooperative Oncology Group \[ECOG\] performance status (PS) \<=2 (Karnofsky \>=60%). * Adequate organ and marrow function as defined below: * Absolute neutrophil count (ANC) \>=1,500/microliter * Platelets \>=100,000/ microliter * Hemoglobin (Hgb) \> 9g/dL (erythrocyte transfusions are allowed to achieve acceptable Hgb) * Total bilirubin within normal limits with the following exceptions: * Participants with tumor involving the liver may have mild to moderate hepatic impairment with total bilirubin \<= 1.5 x upper limit of normal (ULN) * Participants with known Gilbert disease who have serum bilirubin level \<= 1.5 x ULN * Aspartate aminotransferase (AST)/ alanine aminotransferase (ALT) \<=1.5 x institutional ULN ---Participants with tumor involving the liver with AST and ALT \<= 5.0 x ULN and bilirubin \<= 1.5 x ULN may be eligible. * Creatinine clearance (CrCl) \>= 30 mL/min/1.73 m\^2 (glomerular filtration rate \[GFR\] may be used in place of CrCl. Creatinine clearance or eGFR should be calculated per institutional standard) * Creatine phosphokinase (CPK) \<= 2.5 x ULN * Participants with previously treated brain metastases or central nervous system (CNS) metastases are eligible if they have recovered from any acute effects of radiotherapy and not requiring steroids, and any whole brain radiation therapy or any stereotactic radiosurgery was completed at least 2 weeks prior to initiation of therapy. * Human immunodeficiency virus (HIV) positive participants are eligible if on stable dose of highly active antiretroviral therapy (HAART), CD4 counts are \> 350 cells/mm\^3 and viral load is undetectable. * Hepatitis B virus (HBV) positive participants are eligible if they have been treated or are on an appropriate course of antivirals at study entry and with planned monitoring and management according to appropriate guidance including prophylaxis. * Hepatitis C virus (HCV) positive participants are eligible if: * they are on active HCV therapy at study entry or are on an appropriate course of antivirals without documented clinically significant impaired liver function test or hematologic abnormalities and with planned monitoring and management according to appropriate labeling, or if they are post-treatment for HCV; or * they have a negative polymerase chain reaction (PCR). * Systemic corticosteroid therapy (defined as \>= the equivalent of prednisone 10 mg/day) or other immunosuppressive agents such as azathioprine or cyclosporin must be discontinued at least 1 week prior to treatment initiation for recent short course use (\<=14 days) or discontinued at least 4 weeks prior to treatment initiation for long term use (\>14 days). * Contraception requirements as follows: Individuals of child-bearing potential (IOCBP) must agree to use a highly effective method of contraception (e.g., intrauterine device \[IUD\], hormonal, surgical sterilization, abstinence) prior to study entry, for the duration of study participation, and for up to six (6) months after discontinuation of the study drug(s). * Individuals able to father a child must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) for the duration of the study treatment and up to six (6) months after the last dose of the study drug(s). We also will recommend individuals able to father a child with IOCBP partners to ask the partners to be on an effective birth control (hormonal, intrauterine device (IUD), surgical sterilization). Individuals able to father a child must not freeze or donate sperm within the same period. * Nursing participants must be willing to discontinue nursing from study treatment initiation through six (6) weeks after the last dose of the study drug(s). * Participants must be able to understand and willing to sign a written informed consent document. EXCLUSION CRITERIA: * Prior investigational drug, chemotherapy, immunotherapy or any prior radiotherapy (except for palliative bone directed therapy) within the past 14 days prior to the first drug administration. Additionally, FDA-approved hormonal therapy for the treatment or prevention of other malignancies (e.g., breast cancer, prostate cancer) may be continued where in the opinion of the investigator stopping such therapies may increase the risk of disease progression. Potential drug-drug interactions with the hormonal agent will be assessed by the enrolling investigator prior to enrollment. * Participants previously treated with lurbinectedin. * History of anaphylactic allergic reactions attributed to compounds of similar chemical or biologic composition to lurbinectedin or avelumab * Symptomatic or untreated CNS metastases * For Cohort 2: Active autoimmune disease that might deteriorate when receiving avelumab with the exception of: * Diabetes type I, eczema, vitiligo, psoriasis, or hypo- or hyperthyroid diseases not requiring immunosuppressive treatment * Requirement of hormone replacement with corticosteroids if the steroids are administered only for the purpose of hormonal replacement and at doses \<= 10mg of prednisone or equivalent per day * Requirement of steroids for other conditions through a route known to result in a minimal systemic exposure (topical, intranasal, intro-ocular, or inhalation) * Physiologic doses of corticosteroids (\<= the equivalent of prednisone 10 mg/day). The use of corticosteroids as premedication for contrast-enhanced studies is allowed prior to treatment initiation and on study. * Prior organ transplantation including allogenic stem cell transplantation. * Participants who have received or will receive a live vaccine within 30 days prior to the first administration of study intervention. Seasonal flu vaccines that do not contain a live virus and locally authorized/approved COVID-19 vaccines are permitted. * Pregnant people as evaluated by a positive serum or urine beta-human chorionic gonadotropin (beta-hCG) test * Severe uncontrolled intercurrent illness that would limit compliance with study requirements, evaluated by history, physical exam, and chemistry panel.