Testing Mosunetuzumab Alone, With Zanubrutinib, or With Polatuzumab Vedotin for Treating Marginal Zone Lymphoma That Came Back or Didn't Get Better With Treatment

Inclusion Criteria: * Participants must have histologically diagnosed CD20+ marginal zone lymphoma (MZL) as per World Health Organization (WHO) criteria including splenic, nodal, and extranodal subtypes, but excluding gastrointestinal-only marginal zone lymphoma (MZL) with disease assessments that can only be evaluated through endoscopic methods and cutaneous-only MZL. * NOTE: A repeat biopsy to confirm MZL diagnosis is NOT required at time of relapse unless: * The participant has received prior CD3/CD20 bispecific antibody, then a repeat biopsy to document continued CD20+ MZL disease is required after the completion of the prior CD3/CD20 bispecific antibody therapy and prior to study registration. * The participant has splenic MZL in which a bone marrow biopsy pre-registration is required within 42 days prior to registration * Participants must have measurable disease by PET-CT (preferred), or CT as defined by extranodal lesion ≥ 1cm or nodal lesion ≥ 1.5cm. * Participants with splenic MZL are included in the study if spleen standardized uptake value (SUV) (or any splenic masses) is \> liver (standardized uptake value) SUV background and/or spleen size is \> 13cm. * Participants must have staging imaging performed within 42 days prior to registration, as follows. PET-CT baseline scans are preferred. If a baseline PET-CT scan cannot be obtained, CT scans of the neck, chest, abdomen, and pelvis, are acceptable. All disease must be assessed and documented on the Baseline Tumor Assessment Form * Participants must have one or more of the following criteria for further systemic therapy as per the discretion of the treating physician: * Symptoms due to progressive or bulky nodal disease. * Progressive disease that is currently compromising or may compromise normal organ function if left untreated. * Presence of systemic B symptoms (i.e. fevers, weight loss, night sweats). * Presence of symptomatic extranodal disease. * Cytopenias due to bone marrow infiltration or hypersplenism. * An increase in the tempo of disease progression * Participants must not have known or clinically suspected transformation to diffuse large B-cell lymphoma or high-grade B-cell lymphoma. Participants with prior transformed disease but now in relapse with MZL only are allowed on study * Participants with central nervous system (CNS) involvement are eligible if follow-up CNS evaluation shows no evidence of progression, or if the treating physician determines that immediate CNS specific treatment is not required and is unlikely to be required during the first eight cycles (6 months) of protocol therapy * Participants must have relapsed/refractory MZL after at least one line of prior CD20-directed systemic therapy (either as monotherapy or in combination with chemotherapy or lenalidomide). * Prior antibiotic and radiation treatments for localized MZL disease are allowed and do not count as one line of systemic therapy * Participants who have been treated with prior Bruton's tyrosine kinase inhibitor (BTKi) or CD3/CD20 targeting bispecific antibodies for their MZL must have completed treatment 180 days prior to registration and must have received a best response of either a partial or complete response * Participants being treated with strong and moderate CYP3A4 inducers must be off these therapies within 14 days or 5 half-lives of the drug prior to registration, whichever is shorter * Participants must have recovered (\< grade 2) from any side effects of prior therapy, except for alopecia and lymphopenia * Participants must not have been treated with prior polatuzumab vedotin for any condition * Participants must not have received chimeric antigen receptor T-cells (CAR-T) within 28 days prior to registration * Participants must not have received autologous stem cell transplantation within 100 days prior to registration * Participants must not have received allogeneic stem cell transplantation within 180 days prior to registration nor have active graft versus host disease requiring the current use of systemic steroid treatment ≥ 10mg of prednisone (or equivalent) * Participants must not have a condition requiring systemic treatment with either corticosteroids (defined as equivalent to ≥ 10mg prednisone) or other immunosuppressive medications within 7 days prior to registration. * NOTE: Replacement steroid therapy for adrenal or pituitary insufficiency or short-term use of corticosteroids for premedication or treatment of an allergy or hypersensitivity is permitted * Participants must not have known clinically active post-transplant lymphoproliferative disorder * Participants must not have a known history of severe allergic reaction attributed to compounds of similar chemical or biologic composition to mosunetuzumab SQ, zanubrutinib or polatuzumab vedotin * Participants must not have received either primary or booster vaccination with live or attenuated vaccines within 28 days prior to registration * Participants must be ≥ 18 years old at the time of registration * Participants must have Zubrod Performance Status of 0-2 * Participants must have a complete medical history and physical exam within 28 days prior to registration * Absolute neutrophil count ≥ 1.0 x 10\^3/uL (within 28 days prior to registration) * Note: Participants with documented MZL bone marrow involvement or a known/documented Fy (A-/B-) immunophenotype by completed duffy antigen phenotyping (i.e. "Duffy-Null") must have absolute neutrophil count (ANC) ≥ 0.5 x 10\^3/uL. Growth factor use is allowed * Platelets ≥ 75 x 10\^3/uL (within 28 days prior to registration) * Note: Participants with documented MZL bone marrow or splenic involvement must have platelets ≥ 50 x 10\^3/uL * Total bilirubin \< 1.5 x institutional upper limit of normal (ULN) (within 28 days prior to registration) * Note: Participants with history of Gilbert's disease must have total bilirubin ≤ 5 x institutional ULN * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) ≤ 3 × institutional ULN (within 28 days prior to registration) * Participants must have a calculated creatinine clearance ≥ 30 mL/min using the following Cockcroft-Gault Formula. This specimen must have been drawn and processed within 28 days prior to registration. For creatinine clearance formula see the tools on the Clinical Research Associate (CRA) Workbench * Participants must have an international normalized ratio (INR) \< 2 x ULN within 28 days prior to registration * Participants with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, must have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification within 28 days prior to registration. To be eligible for this trial, participants must be class 2B or better, in the opinion of the treating physician * Participants with a known history of human immunodeficiency virus (HIV)-infection must be on effective anti-retroviral therapy at registration and have undetectable viral load test on the most recent test results obtained within 180 days prior to registration * Participants with a known history of chronic hepatitis B virus (HBV) infection must have undetectable HBV viral load while on suppressive therapy on the most recent test results obtained within 180 days prior to registration, if indicated. Participants with a positive hepatitis (Hep) B core antibody are at high risk for reactivation and should receive prophylactic antiviral therapy (e.g., entecavir) before initiation of and throughout the duration of protocol treatment * Participants with a known history of hepatitis C virus (HCV) infection must have been treated and cured. Participants currently being treated for HCV infection must have undetectable HCV viral load test on the most recent test results obtained within 180 days prior to registration, if indicated * Participants must not have any known uncontrolled intercurrent illness (in the opinion of the treating physician) that would jeopardize the participant's safety such as infection, autoimmune conditions, cardiac arrhythmias, angina pectoris, peripheral neuropathy, hypertension and gastrointestinal disorders affecting swallowing and/or absorption of pills * Participants must not require or be receiving anticoagulation with warfarin or equivalent vitamin K antagonists (e.g. phenprocoumon) within 7 days prior to registration * Participants must not have a history of severe bleeding disorder such as hemophilia A, hemophilia B, von Willebrand disease, or history of spontaneous bleeding requiring blood transfusion or other medical intervention * Participants must not have a history of stroke or intracranial hemorrhage within 180 days prior to registration * Participants must not have a history of progressive multifocal leukoencephalopathy * Participants must not have uncontrolled AIHA (autoimmune hemolytic anemia) or ITP (idiopathic thrombocytopenia purpura) * Participants must not have a prior or concurrent malignancy whose natural history or treatment (in the opinion of the treating physician) has the potential to interfere with the safety or efficacy assessment of the investigational regimen * Participants must not be pregnant or nursing (nursing includes breast milk fed to an infant by any means, including from the breast, milk expressed by hand, or pumped). Participants who are of reproductive potential must have agreed to use an effective contraceptive method with details provided as part of the consent process. A person who has had menses at any time in the preceding 12 consecutive months or who has semen likely to contain sperm is considered to be of "reproductive potential." In addition to routine contraceptive methods, "effective contraception" also includes refraining from sexual activity that might result in pregnancy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) including hysterectomy, bilateral oophorectomy, bilateral tubal ligation/occlusion, and vasectomy with testing showing no sperm in the semen * Participants must be offered the opportunity to participate in specimen banking * Participants who can complete the PRO-CTCAE questionnaires in English or Spanish must be offered the opportunity to participate in the patient-reported outcome study * NOTE: As a part of the Oncology Patient Enrollment Network (OPEN) registration process the treating institution's identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system. * Participants must be informed of the investigational nature of this study and must sign and give informed consent in accordance with institutional and federal guidelines. * For participants with impaired decision-making capabilities, legally authorized representatives may sign and give informed consent on behalf of study participants in accordance with applicable federal, local, and Central Institutional Review Board (CIRB) regulations