Ixazomib Citrate and Rituximab in Treating Patients With Indolent B-cell Non-Hodgkin Lymphoma

Inclusion Criteria: * Voluntary written consent must be given before performance of any study related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care * Female patients who: * Are postmenopausal for at least 1 year before the screening visit, OR * Are surgically sterile, OR * If they are of childbearing potential, agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent form through 90 days after the last dose of study drug, OR * Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject; (periodic abstinence \[e.g., calendar, ovulation, symptothermal, post-ovulation methods\] and withdrawal are not acceptable methods of contraception) * Male patients, even if surgically sterilized (i.e., status post-vasectomy), must agree to one of the following: * Agree to practice effective barrier contraception during the entire study treatment period and through 90 days after the last dose of study drug, OR * Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject; (periodic abstinence \[e.g., calendar, ovulation, symptothermal, post-ovulation methods\] and withdrawal are not acceptable methods of contraception) * Patients must have a diagnosis of one of the following B-NHL malignancies: chronic lymphocytic leukemia (CLL)/ small lymphocytic lymphoma (SLL), follicular lymphoma (FL), mantle cell lymphoma (MCL), marginal zone lymphoma (MZL), or Waldenstrom macroglobulinemia (WM)/ lymphoplasmacytic lymphoma (LPL); patients with mucosa associated lymphoid tissue (MALT) subtype of MZL may have relapsed or refractory disease after a course of antibiotic therapy; otherwise, patients will not have received standard systemic treatment for their B-NHL before the time of study enrollment; standard systemic therapy is defined by including any of the following agents, representing a comprehensive list of recommended front-line agents used in the treatment of B-NHL: cytotoxic chemotherapies (bendamustine, cyclophosphamide, doxorubicin, vincristine, chlorambucil, cytarabine, gemcitabine, platinum drugs, etoposide); anti-CD20 antibodies (obinutuzumab, ofatumumab, rituximab); lenalidomide; ibritumomab tiuxetan; proteasome inhibitors (bortezomib, carfilzomib); tyrosine kinase inhibitors (ibrutinib, acalabrutinib, idelalisib); alemtuzumab; corticosteroids unless given for an indication other than treating the B-NHL; or other therapy as determined by the principal investigator (PI) * Disease: CLL/SLL; Criteria for diagnosis: histopathologic or flow cytometric confirmation * Disease: FL; Criteria for diagnosis: histopathologic confirmation * Disease: MZL; Criteria for diagnosis: histopathologic confirmation * Disease: MCL; Criteria for diagnosis: histopathologic confirmation * Disease: WM/LPL; Criteria for diagnosis: Per World Health Organization (WHO) criteria * Eastern Cooperative Oncology Group (ECOG) performance status and/or other performance status 0, 1, or 2 * Absolute neutrophil count (ANC) \>= 1,000/mm\^3 or \>= 500/mm\^3 if neutropenia is attributed to B-NHL (involvement of bone marrow) without growth factor support * Platelet count \>= 100,000/mm\^3 or \>= 75,000/mm\^3 if thrombocytopenia is attributed to B-NHL (involvement of bone marrow or due to splenomegaly or immune thrombocytopenic purpura); platelet transfusions to help patients meet eligibility criteria are not allowed within 3 days before study enrollment * Total bilirubin =\< 1.5 x the upper limit of the normal range (ULN) * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 x ULN * Calculated creatinine clearance \>= 30 mL/min * Patients are required to meet criteria for initiation of therapy for their B-NHL according to published guidelines by the National Comprehensive Cancer Network (NCCN) * Patients must have measurable disease defined by at least one of the following criteria: * Lesions greater than 1.5 cm that can be accurately measured in two dimensions by computed tomography (CT) (preferred), or magnetic resonance imaging (MRI), and are not included in any prior field of radiation given to treat B-NHL * In patients with CLL, circulating lymphocytes \>= 5,000 / mm\^3 * In patients with WM/LPL, measurable serum monoclonal immunoglobulin M (IgM) Exclusion Criteria: * Female patients who are lactating or have a positive serum pregnancy test during the screening period * Major surgery within 14 days before enrollment * Known central nervous system involvement * Infection requiring systemic antibiotic therapy or other serious infection within 14 days before study enrollment * Evidence of current uncontrolled cardiovascular conditions, including uncontrolled hypertension, cardiac arrhythmias, or congestive heart failure, and unstable angina or myocardial infarction within the past 6 months * Systemic treatment, within 14 days before the first dose of ixazomib, with strong inhibitors of cytochrome P450, family 1, subfamily A, polypeptide 2 (CYP1A2) (fluvoxamine, enoxacin, ciprofloxacin), strong inhibitors of cytochrome P450, family 3, subfamily A (CYP3A) (clarithromycin, telithromycin, itraconazole, voriconazole, ketoconazole, nefazodone, posaconazole) or strong CYP3A inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital), or use of ginkgo biloba or St. John's wort * Known ongoing or known active systemic infection, known active hepatitis B or C virus infection, or known human immunodeficiency virus (HIV) positive * Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol * Known allergy to ixazomib, its analogues, or excipients in the various formulations of ixazomib * Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of ixazomib including difficulty swallowing * Diagnosed or treated for another malignancy within 2 years before study enrollment or previously diagnosed (\> 2 years before study enrollment) with another malignancy and have any evidence of residual disease that is symptomatic or requiring treatment; (this may be waived at the discretion of the principal investigator for patients in complete remission if they have not received systemic therapy); patients with non-melanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection * Patient has \>= grade 2 peripheral neuropathy, or grade 1 with pain on clinical examination during the screening period * Participation in other clinical trials with other investigational agents not included in this trial, within 21 days of the start of this trial and throughout the duration of this trial * Patients may not have impending organ compromise from disease as assessed by their treating physician * Prior treatment of B-NHL with radiation therapy, non-standard systemic therapy, or antibiotics (in cases of MZL) within 21 days of the first dose of ixazomib