Experimental CAR T-Cell combo aims to stop lymphoma relapse after transplant

Copied word for word from the study's registry entry, so the wording is the study team's rather than ours.

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative. * Assent, when appropriate, will be obtained per institutional guidelines * Agreement to allow the use of archival tissue from diagnostic tumor biopsies. * If unavailable, exceptions may be granted with study PI approval * Note: For research participants who do not speak English, a short form consent may be used with a City of Hope (COH) certified interpreter/translator to proceed with screening and leukapheresis, while the request for a translated full consent is processed * Age \>= 18 years * Karnofsky performance status (KPS) \>= 70 * Life expectancy \>= 16 weeks at the time of enrollment * Patients with an indication to be considered for HSCT, who are diagnosed with intermediate or high-grade B cell NHL (e.g., diffuse large B-cell lymphoma \[DLBCL\], mantle cell lymphoma \[MCL\], or transformed NHL) in first relapse after achieving complete remission (CR) or did not achieve CR after a first line therapy * Note: COH pathology review should confirm that research participant's diagnostic material is consistent with history of intermediate or high-grade CD19+ malignancy * No known contraindications to myeloablative HSCT, leukapheresis, steroids or tocilizumab, smallpox vaccine and any other modified vaccinia Ankara (MVA)-based vaccines * Patient must be CMV seropositive * Total serum bilirubin =\< 2.0 mg/dL * Participants with Gilbert syndrome may be included if their total bilirubin is =\< 3.0 * Aspartate aminotransferase (AST) \< 2.5 x upper limits of normal (ULN) * Alanine aminotransferase (ALT) \< 2.5 x ULN * Serum creatinine =\< 2.5 x ULN or estimated creatinine clearance of \>= 40 mL/min per the Cockcroft-Gault formula, and the participant is not on hemodialysis * Absolute neutrophil count \>= 1000/uL (Transfusions and growth factors must not be used to meet this requirement at initial screening) * Hemoglobin (Hb) \>= 8 g/dl (Transfusions and growth factors must not be used to meet this requirement at initial screening) * Platelet count \>= 50,000/uL (\>= 30,000/uL if bone marrow plasma cells are =\> 50 percent of cellularity) (Transfusions and growth factors must not be used to meet this requirement at initial screening) * Left ventricular ejection fraction \>= 45 percent within 8 weeks before enrollment * Oxygen (O2) saturation \> 92% without requiring supplemental oxygen * Women of childbearing potential (WOCBP): negative urine or serum pregnancy test * If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required * Agreement by females and males of childbearing potential to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 6 months after the last dose of protocol therapy. * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only) Exclusion Criteria: * Prior autologous/allogeneic stem cell transplant * Growth factors within 14 days of enrollment * Platelet transfusions within 7 days of enrollment * Concurrent use of systemic steroids or chronic use of immunosuppressant medications. Recent or current use of inhaled or topical steroids in standard doses is not exclusionary. Physiologic replacement of steroids (prednisone =\< 5 mg /day, or equivalent doses of other corticosteroids) is allowed * Patients with active autoimmune disease requiring systemic immune suppressive therapy are not allowed * Participants may not be receiving any other investigational agents or concurrent biological therapy, chemotherapy, or radiation therapy * Any standard contraindications to myeloablative HSCT per standard of care practices at COH * Subjects with clinically significant arrhythmia or arrhythmias not stable on medical management within two weeks of screening * Subjects with a known history or prior diagnosis of optic neuritis or other immunologic or inflammatory disease affecting the central nervous system (CNS), including seizure disorder, any measurable masses of CNS, or any other active CNS disease. Note: Research participants with a history of CNS disease that has been effectively treated to complete remission (\< 5 white blood cells \[WBC\] / mm\^3 and no blasts in cerebrospinal fluid \[CSF\]) will be eligible * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agents or cetuximab * Known bleeding disorders (e.g., von Willebrand's disease) or hemophilia * History of stroke or intracranial hemorrhage within 6 months prior to screening * History of other malignancies, except for malignancy surgically resected (or treated with other modalities) with curative intent, basal cell carcinoma of the skin or localized squamous cell carcinoma of the skin; non-muscle invasive bladder cancer; malignancy treated with curative intent with no known active disease present for \>= 3 years. * Clinically significant uncontrolled illness * Active infection requiring antibiotics * Immunodeficiency virus (human immunodeficiency virus \[HIV\]) positive * Active viral hepatitis * Females only: Pregnant or breastfeeding * Any other condition that would, in the investigator's judgment, contraindicate the subject's participation in the clinical study due to safety concerns with clinical study procedures * Procedures (including compliance issues related to feasibility/logistics). Prospective participants who, in the opinion of the investigator, may not be able to comply with all study * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)