Acalabrutinib and Anti-CD19 CAR T-cell Therapy for the Treatment of B-cell Lymphoma

Inclusion Criteria: * Histologically confirmed large B-cell lymphoma, including diffuse large B-cell lymphoma (DLBCL) not otherwise specified, primary mediastinal large B-cell lymphoma, high grade B-cell lymphoma, and DLBCL arising from follicular lymphoma, and indolent (grade 1-3a) FL * Criteria must be met for receiving commercial axi-cel per Food and Drug Administration (FDA) label * \>= 18 years of age * Patients must be capable of understanding and providing a written informed consent * Negative serum pregnancy test within 2 days of initiating acalabrutinib for women of childbearing potential (WOCBP), defined as those who have not been surgically sterilized or who have not been free of menses for at least 1 year * Fertile male and WOCBP patients must be willing to use highly effective contraceptive methods before, during, and for at least 4 months after the CAR T-cell infusion or within 2 days of acalabrutinib, whichever is longer * Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 * Creatine clearance (CrCl) \> 50 mL/min or serum creatinine =\< 2.5 * Total bilirubin =\< 1.5x the upper limit of normal * Adequate pulmonary function, defined as =\< grade 1 dyspnea and oxygen saturation (SaO2) \>= 92% on room air * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 3x the upper limit of normal * Adequate cardiac function, defined as left ventricular ejection fraction (LVEF) of \>= 50% and without evidence for pericardial effusion * At least 1 measurable lesion \>= 15 mm according to the International Working Group consensus response evaluation criteria in lymphoma (Younes 2017) * HIV POSITIVE COHORT: Human immunodeficiency virus (HIV)-1 or HIV-2 infection, as documented by any federally approved, licensed HIV test * HIV POSITIVE COHORT: HIV plasma HIV-1 ribonucleic acid (RNA) below detected limit obtained by Food and Drug Administration (FDA)-approved assays within 4 weeks prior to registration * HIV POSITIVE COHORT: CD4 cell count greater than 200 cells/mm3 obtained within 2 weeks prior to enrollment at any U.S. laboratory that has a clinical laboratory improvement amendments (CLIA) certification or its equivalent * HIV POSITIVE COHORT: Anti-retroviral treatment (ART) should be initiated \> 4 weeks prior to study drug so that toxicity assessment of ART is separated from study drug. If patient is on an ART regimen that contains a strong CYP3A inhibitor (e.g ritonavir and cobicistat) or CYP3A inducer (e.g. efavirenz), changes in ART therapy should be considered in collaboration with HIV provider * HIV POSITIVE COHORT: No acute active HIV-associated opportunistic infection requiring antibiotic treatment * HIV POSITIVE COHORT: No uncontrolled systemic fungal, bacterial, viral, or other infection * HIV POSITIVE COHORT: Hemoglobin \> 8.0 g/dl * HIV POSITIVE COHORT: Serum creatinine \< 1.5 mg/dL OR creatinine clearance \> 60 mL/min AST and ALT \< 2.5 x ULN Exclusion Criteria: * Active and uncontrolled systemic or clinically significant infection that would contraindicate myelosuppressive therapy or CART infusion * Patients intolerant of acalabrutinib * Requires treatment with proton pump inhibitors (eg, omeprazole, esomeprazole, lansoprazole, dexlansoprazole, rabeprazole, or pantoprazole). Note: Subjects receiving proton pump inhibitors who switch to H2-receptor antagonists or antacids are eligible for enrollment to this study * Patients with detectable cerebrospinal fluid malignant cells, or brain metastases, or with a history of cerebrospinal fluid malignant cells or brain metastases * History of a seizure disorder, cerebrovascular ischemia/hemorrhage, dementia, cerebellar disease, or any autoimmune disease with central nervous system (CNS) involvement * Use of a strong CYP3A inhibitor OR inducer within 7 days of starting study drugs or requirement of use of strong CYP3A inhibitor OR inducer at the time of enrollment * Disease that is known to be refractory to BTK inhibition * Absolute neutrophil count (ANC) \< 1000/ul * Platelets \< 50K/ul * Another active malignancy requiring systemic treatment, unless approved by principal investigator (PI) * Clinically significant cardiovascular disease such as uncontrolled or symptomatic arrhythmias, congestive heart failure, or myocardial infarction within 6 months of screening, or any class 3 or 4 cardiac disease as defined by the New York Heart Association Functional Classification. Subjects with controlled, asymptomatic atrial fibrillation during screening can enroll on study * Inability to swallow whole pills, malabsorption syndrome, disease significantly affecting gastrointestinal function, or resection of the stomach or small bowel that is likely to affect absorption, symptomatic inflammatory bowel disease, partial or complete bowel obstruction, or gastric restrictions and bariatric surgery, such as gastric bypass * Active bleeding, history of bleeding diathesis (eg, hemophilia or von Willebrand disease) * Uncontrolled AIHA (autoimmune hemolytic anemia) or ITP (idiopathic thrombocytopenic purpura) * Receiving anticoagulation with warfarin or equivalent vitamin K antagonists (eg, phenprocoumon) within 7 days of first dose of study drug * Prothrombin time/international normalized ratio (INR) or activated partial thromboplastin time (aPTT) (in the absence of Lupus anticoagulant) \> 2 x upper limit of normal (ULN) * History of significant cerebrovascular disease or event, including stroke or intracranial hemorrhage, within 6 months before the first dose of study drug * Major surgical procedure within 7 days of first dose of study drug. Note: If a subject had major surgery, they must have recovered adequately from any toxicity and/or complications from the intervention before the first dose of study drug * Hepatitis B or C serologic status: subjects who are hepatitis B core antibody (anti-HBc) positive and who are surface antigen negative will need to have a negative polymerase chain reaction (PCR). Those who are hepatitis B surface antigen (HbsAg) positive or hepatitis B PCR positive will be excluded * Subjects who are hepatitis C antibody positive will need to have a negative PCR result. Those who are hepatitis C PCR positive will be excluded * Pregnant or breast feeding