CD19/CD22 Bicistronic Chimeric Antigen Receptor (CAR) T Cells in Children and Young Adults With Recurrent or Refractory CD19/CD22-expressing B Cell Malignancies

Inclusion Criteria

• - INCLUSION CRITERIA: - Diagnosis - Participant must: - Have pathology confirmed B cell ALL (not isolated to the testis or CNS), CML with ALL transformation, or high-grade lymphoma (e.g., Burkitt's lymphoma, B-lymphoblastic lymphoma, diffuse large B-cell lymphoma, inclusive of low-grade lymphoma that has transformed to high grade disease); and - Have relapsed or been refractory after at least one standard chemotherapy regimen and at least one salvage treatment. Participants with Philadelphia chromosome + ALL must have failed prior tyrosine kinase inhibitor; and - Be ineligible for allogeneic stem cell transplant (SCT), have refused SCT, or have recurred after SCT; and - Have no evidence of graft-versus- host disease (GVHD) and have been without immunosuppressive agents for at least 30 days prior to apheresis, if undergone prior allogeneic SCT; and - Be unable to access (in a timely manner), ineligible for, or have relapsed/failed after or not responded to a commercially available CD19 CAR T-cell construct; and - Have evidence of at least minimal residual disease or PET-avid disease (lymphoma) at the time of enrollment. - CD22/CD19 expression - CD19 must be detected on >15% of the malignant cells by immunohistochemistry or > 80% by flow cytometry. - CD22 positivity must be confirmed. - Age >= 3 years of age and <=39 years of age at time of enrollment. - Clinical Performance status: Participants >= 16 years of age: Karnofsky >= 50%; Participants < 16 years of age: Lansky scale >= 50%. - Participants must have adequate organ and marrow function as defined below: - leukocytes >= 750/mcL* - platelets >= 50,000/mcL* - total bilirubin <=2 X ULN (except in the case of participants with documented Gilbert's disease > 3x ULN) - AST(SGOT)/ALT(SGPT) <=10 X institutional upper limit of normal - creatinine <= the maximum for age listed in the table below OR - measured creatinine clearance >=60 mL/min/1.73 m^2 for participants with creatinine levels above the max listed below per age. - Age (Years) <= 5 / Maximum Serum Creatinine (mg/dL) <= 0.8 - Age (Years) 6 to <= 10 / Maximum Serum Creatinine (mg/dL) <= 1.0 - Age (Years) >10 / Maximum Serum Creatinine (mg/dL) <= 1.2 - a participant will not be excluded because of pancytopenia >= Grade 3 if it is due to underlying bone marrow involvement by leukemia - Central nervous system (CNS) Status - Participants with leukemia with CNS 1 and 2 disease are eligible in the absence of exclusion criteria - Participants of child-bearing or child-fathering potential must be willing to practice effective birth control from the time of enrollment until 12 months following completion of study treatment for women and for 4 months following completion of study treatment for men. - Participants who are breastfeeding or plan to breastfeed must agree to discontinue/postpone breastfeeding while on study therapy and until 1 month after the administration of CAR. - Cardiac function: Left ventricular ejection fraction >= 45% or fractional shortening >=28% - Pulmonary Function - Baseline oxygen saturation >92% on room air at rest - Ability of participant or Legally Authorized Representative (LAR) to understand and the willingness to sign a written informed consent document. - Ability and willingness of participant or Legally Authorized Representative (LAR) to co- enroll on 15-C-0028: Follow-up Evaluation for Gene-Therapy Related Delayed Adverse Events after Participation in Pediatric Oncology Branch Clinical Trials. EXCLUSION CRITERIA: Participants meeting any of the following criteria are not eligible for participation in the study: - Participants with CNS3 disease, progressing neurologic signs* of CNS disease, radiologically detected active CNS lymphoma (*resolving manifestation or persistent and/or irreversible findings from prior CNS involvement (e.g., blindness) is not exclusionary) - Hyperleukocytosis (>= 50,000 blasts/microL) - Positive serum or urine beta-HCG pregnancy test performed at screening. - Participants will be excluded based on prior therapy if they fail to meet following washout criteria: - Therapy: Systemic Chemotherapy, anti-neoplastic agents, antibody- based therapies - Washout*: >=2 weeks - Exceptions: 6 weeks for clofarabine or nitrosoureas; No washout for prior intrathecal chemotherapy, steroid therapy, hydroxyurea (no dose increases within prior 2 weeks) or ALL maintenance-type chemotherapy (vincristine, 6-mercaptopurine, oral methotrexate, or a tyrosine kinase inhibitor for participants with Ph+ ALL) provided there is recovery from any acute toxic effects - Therapy: Radiation - Washout*: >=3 weeks - Exceptions: No time restriction with radiation therapy if the volume of bone marrow treated is less than 10% and the participant has measurable/evaluable disease outside the radiation window - Therapy: Allogeneic Stem Cell Transplant - Washout*: >= 100 days since SCT; >= 30 days since completion of immunosuppression; >= 6 weeks since donor lymphocyte infusion (DLI) - Exceptions: Cannot have evidence of active graft-versus-host disease (GVHD) - Therapy: CAR T-Cell Therapy or other Adoptive Cell Therapy - Washout*: > 30 days post infusion - Washout: Time between therapy and apheresis - Positive HIV antibodies consistent with active HIV. - Positive hepatitis C antibodies or positive Hepatitis B surface antigen (HbsAG) indicative of current/active HCV/HBV. - Active second malignancy other than in situ carcinoma of the cervix, unless the tumor was treated with curative intent at least two years previously and participant is in remission. - History of severe, immediate hypersensitivity reaction attributed to compounds of similar chemical or biologic composition to any agents used in study or in the manufacturing of the cells. - Uncontrolled, symptomatic, intercurrent illness or social situations that would limit compliance with study requirements or in the opinion of the PI would pose an unacceptable risk to the participant.