Acalabrutinib With or Without Obinutuzumab in Treating Patients With Early-Stage Chronic Lymphocytic Leukemia or Small Lymphocytic Lymphoma

Inclusion Criteria: * Age \>= 18 years * Diagnosis of: * Biopsy-proven small lymphocytic lymphoma (SLL) , or * Diagnosis of chronic lymphocytic leukemia (CLL) with a clonal B-cell population in the peripheral blood with immunophenotyping consistent with CLL as follows: * The population of lymphocytes share both B-cell antigens (CD19, CD20 \[typically dim expression\], or CD23) as well as CD5 in the absence of other pan-T-cell markers (CD3, CD2, etc.) * Clonality as evidenced by kappa or lambda light chain expression (typically dim immunoglobulin expression) or other genetic method (e.g. IGHV analysis) * Before diagnosing CLL or SLL, mantle cell lymphoma must be excluded by demonstrating a negative fluorescence in situ hybridization (FISH) analysis for t(11;14)(IgH/CCND1) * Patients must be previously untreated * Note: Prior chemotherapy or monoclonal antibody based therapy for treatment of CLL or SLL will be considered prior therapy; nutraceutical treatments with no established benefit in CLL (such as epigallocatechin gallate or EGCG, found in green tea or other herbal treatments or supplemental vitamins) will not be considered "prior treatment"; prior corticosteroid therapy for an indication other than CLL/SLL will not be considered "prior treatment" * All patients will undergo testing for prognostic factors according to the CLL-IPI (testing obtained =\< 730 days prior to registration) * Note: If the results for any of the prognostic factors included in the CLL-IPI are unknown including IGVH mutation status results not being available due to a failed laboratory assay, the patient is not eligible * Note: When determining CLL-IPI, use most recent test results, if more than one result is available * Note: Patients with CLL-IPI risk category of high risk or very high risk (total score of 4-10) will be randomized to Arms A or B * Note: Patients with CLL-IPI risk category of low risk or intermediate risk (total score of 0-3) will be registered to Arm C * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1, or 2 * Provide written informed consent * Willing to provide blood and saliva samples for correlative research purposes * Willing to return to enrolling institution for follow-up (during the active monitoring phase of the study) * For high risk and very high risk CLL-IPI (Arms A and B) only: Absolute neutrophil count (ANC) \>= 1500/mm\^3 (obtained =\< 30 days prior to randomization) * For high risk and very high risk CLL-IPI (Arms A and B) only: Platelet count \>= 100,000/mm\^3 (obtained =\< 30 days prior to randomization) * For high risk and very high risk CLL-IPI (Arms A and B) only: Hemoglobin \>= 11.0 g/dL (obtained =\< 30 days prior to randomization) * For high risk and very high risk CLL-IPI (Arms A and B) only: Aspartate aminotransferase (aspartate transaminase \[AST\]) =\< 3 x upper limit of normal (ULN) (obtained =\< 30 days prior to randomization) * For high risk and very high risk CLL-IPI (Arms A and B) only: Creatinine =\< 1.5 X ULN (obtained =\< 30 days prior to randomization) * For high risk and very high risk CLL-IPI (Arms A and B) only: Total bilirubin =\< 1.5 x upper limit of normal (ULN) (or total bilirubin =\< 3.0 x ULN with direct bilirubin =\< 1.5 x ULN in patients with well-documented Gilbert's syndrome (obtained =\< 30 days prior to randomization) * For high risk and very high risk CLL-IPI (Arms A and B) only: Prothrombin time (PT), international normalized ratio (INR), and partial thromboplastin time (PTT) =\< 1.5 X ULN OR if patient is receiving anticoagulant therapy and PT or PTT is within therapeutic range of intended use of coagulants (obtained =\< 30 days prior to randomization) * Negative serum pregnancy test done =\< 7 days prior to registration, for persons of childbearing potential only * Will provide bone marrow aspirate sample for correlative research purposes Exclusion Criteria: * Date of CLL/SLL diagnosis \>= 24 months prior to registration * Prior exposure to ibrutinib or to a BCR inhibitor (e.g. Btk or PI3 kinase or Syk inhibitors) or a BCL-2 inhibitor (e.g. venetoclax) * Known central nervous system (CNS) lymphoma or leukemia * Patients with any of the following indications for chemotherapy: * Evidence of progressive marrow failure as manifested by the development of or worsening anemia (=\< 11 g/dL) and/or thrombocytopenia (=\< 100 x 10\^9/L) not due to autoimmune disease * Symptomatic or progressive lymphadenopathy, splenomegaly or hepatomegaly * One or more of the following disease-related symptoms: * Weight loss \>= 10% within the previous 6 months * Extreme fatigue attributed to CLL * Fevers \>= 100.4 degrees Fahrenheit (F) for 2 weeks without evidence of infection * Drenching night sweats without evidence of infection * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Patients known to be human immunodeficiency virus (HIV) positive and currently receiving antiretroviral therapy; NOTE: Patients known to be HIV positive, but without clinical evidence of an immunocompromised state, are eligible for this trial * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm * Other active malignancy =\< 2 years prior to registration; EXCEPTIONS: Non-melanotic skin cancer, carcinoma-in-situ of the cervix, or early stage prostate cancer * History of myocardial infarction =\< 6 months prior to registration, or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias * For high risk and very high risk CLL-IPI (Arms A and B) only: * Any of the following: * Pregnant persons * Nursing persons * Persons of childbearing potential who are unwilling to employ highly effective contraception * Serologic status reflecting active hepatitis B or C infection * NOTE: Subjects with hepatitis B core antibody positive who are surface antigen negative or who are hepatitis C antibody positive will need to have a negative polymerase chain reaction (PCR) result before randomization; those who are hepatitis B surface antigen positive or hepatitis B PCR positive and those who are hepatitis C PCR positive will be excluded * History of stroke or intracranial hemorrhage within 6 months before randomization * History of bleeding diathesis (e.g. hemophilia, von Willebrand disease) * Requires or receiving anticoagulation with warfarin or equivalent vitamin K antagonists (e.g. phenprocoumon) within 7 days of first dose of study drug and while on study * Requires treatment with a strong CYP3A inducer * Presence of a gastrointestinal ulcer diagnosed by endoscopy within 3 months before screening * History of confirmed progressive multifocal leukoencephalopathy (PML) * Received a vaccination with a live vaccine =\< 28 days prior to randomization