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

Background: * Acute lymphoblastic leukemia (ALL) accounts for approximately 25% of childhood cancer. Survival rates have improved, but outcomes for some subgroups, including infants and young adults remain poor, and survival for patients who relapse is \< 50%, despite allogeneic stem cell transplant following second remission. * Cluster of differentiation 19 (CD19 immune escape has been observed by several groups following CD19-chimeric antigen receptor (CAR) therapy for B-ALL. Investigation of this phenomenon reveals a complex biology responsible for loss or downregulation of CD19 expression observed in these cases. * Sequential therapy using cluster of differentiation 22 (CD22)-CARs to treat CD19 dim/lo escape is associated with rapid development of resistance due to CD22 downregulation. This trial will test whether simultaneous targeting of CD19 and CD22 using a novel bivalent CD19/22-CAR is safe and feasible. Objectives: -Assess the safety of administering escalating doses of autologous CD19/CD22-CAR engineered T cells that meet established release specifications in children and young adults with CD19+CD22+ B cell ALL, isolated central nervous system (CNS) ALL, or lymphoma following a cyclophosphamide/fludarabine conditioning regimen. Eligibility: -Participants between \>= 3 years and \<= 39 years of age, with CD19+/CD22+ B cell ALL, isolated CNS 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 who meet standard Phase I eligibility criteria. Design: * Phase I, 3 + 3 dose escalation design using the following dose levels: -1: 1 x 10\^5 transduced T cells/kg (+/- 20%); 1: 3 x 10\^5 transduced T cells/kg (+/- 20%); 2: 1 x 10\^6 transduced T cells/kg; and 3: 3 x 10\^6 transduced T cells/kg (+/- 20%); 4: 1 x 10\^7 transduced T cells/kg (+/- 20%). * Participants will receive a lymphodepleting preparative regimen of fludarabine (25 mg/m\^2/day (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. Participants who are CAR pretreated (with exception for those with an interval hematopoietic stem cell transplantation (HSCT) will receive increased lymphodepleting preparative regimen of fludarabine (30\^mg/m\^2/d x 4 on Days -5, -4, -3, -2) and cyclophosphamide (600 mg/m\^2/d x 2 on Days -3, -2) followed by infusion of CD19/CD22-CAR T-cells on D0. If fludarabine is unavailable, pentostatin may be given as an alternative. * Patients will be evaluated sequentially for toxicity, antitumor effects, CAR expansion and persistence, as well as research correlatives.