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

Background: * Despite improvements in therapy, acute lymphoblastic leukemia (ALL) contributes to significant morbidity and mortality for children and young adults with cancer. CD19-CAR and CD22-CAR therapy have proven highly effective in inducing remission in patients with relapsed/refractory disease. * Immune escape has been observed by several groups following CD19-CAR and CD22- CAR therapy for B-ALL. Investigation of this phenomenon reveals a complex biology responsible for loss or downregulation of CD19 and/or CD22 expression observed in these cases. * The challenges encountered with currently available CD19- and CD22-directed CAR T cells in B-ALL demonstrates the need for combinatorial treatment strategies simultaneously targeting two antigens, such as CD19 and CD22, to enhance the long-term effectiveness of CARs. * We have previously treated patients with B-ALL on a phase 1/2 clinical trial using a bivalent CD19/22 CAR T-cell as a first combinatorial treatment strategy. This CAR T-cell construct is well-tolerated and has yielded responses however there has been limited CAR T cell expansion and persistence. Additionally, the previously tested CD19/CD22 bivalent CAR T-cell construct is limited in its ability to target CD22. * This new CD19/22 targeted construct being tested in this clinical trial has improved dual targeting capability based on preclinical data/evaluation. Objectives: * Phase I: Assess the safety of administering escalating doses of autologous CD19/CD22- CAR engineered T cells in children and young adults with B cell ALL or lymphoma following a cyclophosphamide/fludarabine conditioning regimen. * Phase II: Determine the efficacy of CD19/CD22 therapy in participants with B-ALL/ B-LBL.. Eligibility: -Participants between \>= 3 years and \<= 39 years of age, with CD19+ and/or CD22+ B cell ALL or lymphoma who have relapsed or have refractory disease after at least one standard chemotherapy regimen and one salvage regimen, with no alternative curative options. Design: * Phase I, 3 + 3 dose escalation design across 4 cohorts (B-ALL/B-cell lymphoblastic lymphoma: A: low-disease burden (\<25 % marrow blasts without extramedullary disease) vs. B: high-disease burden (\>= 25 % marrow blasts or with EMD). C: B-cell non-Hodgkin lymphoma D: CD19+ or CD22+ single antigen positivity using the following dose levels: -2: 1 x 10\^5 transduced T cells/kg (+/- 20%); -1: 3 x 10\^5 transduced T cells/kg (+/- 20%); 1: 1 x 10\^6 transduced T cells/kg (+/- 20%); and 2: 3x 10\^6 transduced T cells/kg (+/- 20%). Cohorts will enroll concurrently. * Participants will be treated based on disease burden and will receive 1 of 2 lymphodepleting preparative regimens: * Lymphodepleting preparative regimen # 1: Fludarabine (30 mg/m\^2/d x 3 on Days -4, -3, -2) and cyclophosphamide (900 mg/m\^2/d x 1 on Day -2) followed by infusion of CD19/CD22-CAR T-cells on D0. * Lymphodepleting preparative regimen #2: Fludarabine (30 mg/m2/d x 4 on Days -5, -4, -3, -2) and cyclophosphamide (600 mg/m2/d x 2 on Days -3, -2) followed by infusion of CD19/CD22-CAR T-cells on D0. * Determination for use of LD regimen #1 versus #2 will be based on pre-treatment absolute lymphocyte count, pre-existing cytopenias, receipt of prior CAR T-cell therapy, high disease burden and assessment of infection risk. * Participants will be evaluated sequentially for toxicity, antitumor effects, CAR expansion and persistence, and other biologic correlatives.