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

* 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 * 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 * Cohorts A1b, B1b, C2b * CD19 must be detected on \>15% of the malignant cells by immunohistochemistry or \> 80% by flow cytometry. * CD22 positivity must be confirmed. * Cohorts D1b, 2 B-ALL * CD19 or 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) requiring systemic immunosuppression * 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.