Lenalidomide in Improving Immune Response to Vaccine Therapy in Patients With Chronic Lymphocytic Leukemia, Small Lymphocytic Lymphoma, or Monoclonal B Cell Lymphocytosis

Inclusion Criteria: * Diagnosis of: 1. CLL according to the National Cancer Institute (NCI) criteria 2. Small lymphocytic lymphoma (SLL) according to the World Health Organization (WHO) criteria 3. MBL according to the consensus criteria * This includes previous documentation of: * Biopsy-proven small lymphocytic lymphoma or * Diagnosis of CLL or MBL as evidenced by all of the following: * Clonal B-cell population in the peripheral blood with immunophenotyping consistent with CLL defined as: * The population of lymphocytes share both B-cell antigens (cluster of differentiation \[CD\]19, 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 k (kappa) or lambda light chain expression (typically dim immunoglobulin expression) or other genetic method (e.g. immunoglobulin heavy chain variable \[IGHV\] analysis) NOTE: splenomegaly, hepatomegaly, or lymphadenopathy are not required for the diagnosis of CLL * Patients with a peripheral blood B-cell lymphocyte count of \< 5 x 10\^9/L and no evidence of lymphadenopathy or organomegaly will be classified as MBL; patients with a peripheral blood B-cell lymphocyte count of \< 5 x 10\^9/L who have evidence of lymphadenopathy will be classified as SLL; patients with a peripheral blood B-cell lymphocyte count \>= 5 x 10\^9/L will be considered to have CLL * Before diagnosing MBL, CLL or SLL, mantle cell lymphoma must be excluded by demonstrating a negative fluorescence in situ hybridization (FISH) analysis for t(11;14)(IgH/cyclin D 1 \[CCND1\]) on peripheral blood or tissue biopsy or negative immunohistochemical stains for cyclin D1 on involved tissue biopsy * CLL or SLL patients only (does not apply to MBL patients): Rai stage 0-1 (both CLL and SLL patients can be staged using the Rai system) * Patients must not previously have received the Prevnar 13 pneumococcal vaccination; NOTE: previous vaccination with Pneumovax (PCV23) is permitted but must have been at least 365 days prior to registration * Patients must be previously untreated and must NOT have 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 degree Fahrenheit (o\^F) for 2 weeks without evidence of infection * Drenching night sweats without evidence of infection Note: 1) 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) will not be considered "prior treatment" 2) Prior corticosteroid therapy for an indication other than CLL/SLL will not be considered "prior treatment"; previous use of corticosteroids for treatment of autoimmune complications of CLL/SLL does not constitute prior therapy for CLL/SLL * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * Absolute neutrophil count (ANC) \>= 1000/mm\^3 * Platelet count \>= 100,000/mm\^3 * Hemoglobin \>= 11.0 g/dL * Total bilirubin =\< 1.5 x upper limit of normal (ULN) unless due to Gilbert's disease; if total bilirubin is \> 1.5 x ULN, a direct bilirubin should be performed and must be \< 1.5 mg/dL for Gilbert's to be diagnosed * Aspartate transaminase (AST) =\< 3 x ULN * Calculated creatinine clearance must be \>= 50 ml/min using the Cockcroft-Gault formula * Provide informed written consent * Willing to return to enrolling institution for follow-up (during the active monitoring phase of the study) * All study participants must be willing to be registered into the mandatory Revlimid Risk Evaluation and Mitigation Strategy (REMS)® program, and be willing and able to comply with the requirements of the REMS® program; NOTE: Actual registration in the Revlimid REMS® program may occur after the patient is randomized since this requirement only applies to patients randomized to Arm A * Females of reproductive potential must be willing to adhere to the scheduled pregnancy testing as required in the Revlimid REMS® program; NOTE: This requirement only applies to patients randomized to Arm A * Able to take aspirin (81 or 325 mg) daily as prophylactic anticoagulation (patients intolerant to acetylsalicylic acid \[ASA\] may use warfarin or low molecular weight heparin) * Willing to provide blood samples for correlative research purposes Exclusion Criteria: * Palpable lymph nodes \> 3 cm in maximal dimension * Any of the following: * Pregnant women * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception * Any of the following comorbid conditions: * New York Heart Association classification III or IV cardiovascular disease * Recent myocardial infarction (=\< 30 days) * Uncontrolled infection * Immunocompromised patients and 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 * Other active primary malignancy requiring treatment or limiting survival to =\< 2 years prior to registration * Any radiation therapy =\< 28 days prior to registration * Any major surgery =\< 28 days prior to registration * Current use of corticosteroids; EXCEPTION: low doses of steroids (=\< 10 mg of prednisone or equivalent dose of other steroid) used for treatment of nonhematologic medical conditions; NOTE: previous use of corticosteroids is allowed * Active hemolytic anemia requiring immunosuppressive therapy or other pharmacologic treatment; NOTE: patients who have a positive Coombs test but no evidence of hemolysis are NOT excluded from participation * History of deep venous thromboses or pulmonary embolism =\< 365 days prior to registration * Co-existent diffuse large B-cell lymphoma (Richter's transformation)