CD33-CAR T Cell Therapy for the Treatment of Recurrent or Refractory Acute Myeloid Leukemia

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative * Assent, when appropriate, will be obtained per institutional guidelines * 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, while the request for a translated full consent is processed * Agreement to allow the use of archival tissue from diagnostic tumor biopsies * If unavailable, exceptions may be granted with Study principal investigator (PI) approval * Age: \>= 18 years * Karnofsky Performance Scale (KPS) \>= 70 * Life expectancy \>= 16 weeks at the time of enrollment * Prior allogeneic transplant allowed if \> 6 months prior to study enrollment * Participant must have a confirmed diagnosis of active CD33+ AML de novo, or secondary OR participants who are at a high risk for disease recurrence * Relapsed AML is defined as patients that had a first complete response (CR) before developing recurrent disease (increased bone marrow blasts) * Refractory AML is defined as patients that have not achieved a first CR after induction chemotherapy. For patients with AML evolving from myelodysplastic syndrome, they should have completed at least one cycle of induction chemotherapy * Research participants must have bone marrow and/or peripheral blood samples available for confirmation of diagnosis of AML * CD33 positivity must be confirmed by either flow cytometry or immunohistochemistry within 90 days of study entry. Cytogenetics, flow cytometry, and molecular studies (such as FLT-3 status) will be obtained as per standard practice * Research participants who are at a high risk of disease recurrence, they must have historical bone marrow and/or peripheral blood samples available for confirmation of diagnosis of AML * No known contraindications to lymphodepleting agents, steroids, tocilizumab and/or cetuximab, or the investigational agent * Total serum bilirubin =\< 2.0 mg/dL * Participants with Gilbert syndrome may be included if their total bilirubin is =\< 3.0 * Aspartate aminotransferase (AST) =\< 3 x the upper limit of normal (ULN) * Alanine aminotransferase (ALT) =\< 3 x ULN * Estimated creatinine clearance of \>= 60 mL/min per the Cockcroft-Gault formula, and the participant is not on hemodialysis * Left ventricular ejection fraction \>= 50% within 8 weeks before enrollment * Oxygen (O2) saturation \> 92% not requiring oxygen supplementation * 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) * Research participants must have a potential donor or stem cell source identified for allogeneic transplantation, either related (7/8 or 8/8 allele matched or haploidentical) * DONOR: The identified donor must be the original donor whose stem cells were used for the research participant's allogeneic hematopoietic stem cell transplantation (alloSCT) * DONOR: The donor must be HIV negative * DONOR: KPS \>= 70 * DONOR: Documented body weight Exclusion Criteria: * Prior allogeneic transplant if \< 6 months prior to enrollment * Concurrent use of systemic steroids or chronic use of immunosuppressant medications should be stopped 28-days prior to enrollment. Recent or current use of inhaled or topical steroids in standard doses is not exclusionary. Physiologic replacement of steroids (prednisone =\< 7.5 mg/day, or equivalent doses of other corticosteroids) is allowed * Participants with active autoimmune disease, including graft versus host disease (GvHD), requiring systemic immune suppressive should be stopped 28-days prior to enrollment * Participants may not be receiving any other investigational agents and are not dependent on concurrent biological therapy, chemotherapy, or radiation therapy * With exception to Hydrea which must be stopped prior to initiation of lymphodepletion * Research participants on active systemic antifungal treatment within 8 weeks of enrollment are not eligible. However, participants on antifungal prophylaxis are eligible * Not applicable at the time of enrollment if the research participant's donor is undergoing leukapheresis * Subjects with \>= Grade 2 myelofibrosis on bone marrow biopsy * Subjects with clinically significant arrhythmia or arrhythmias not stable on medical management within two weeks of screening if the patient is undergoing leukapheresis. Patients with controlled atrial arrythmia is allowed * Known bleeding disorders (e.g., von Willebrand's disease) or hemophilia * History of stroke or intracranial hemorrhage within 6 months prior to screening * Subjects with presence of other active malignancy, however, research participants with history of prior malignancy treated with curative intent and in complete remission are eligible * Clinically significant uncontrolled illness * Active infection requiring antibiotics * Research participants who have tested human immunodeficiency virus (HIV) positive, or have active hepatitis B or C infection based on testing performed within 4 weeks of enrollment * 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 * 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)