This is a phase I, open-label, non-randomized study that will enroll pediatric and young adult research participants with relapsed or refractory non-CNS solid tumors to evaluate the safety, feasibility, and efficacy of administering T cell products derived from the research participant's blood that have been genetically modified to express a B7H3-specific receptor (chimeric antigen receptor, or CAR) that will target and kill solid tumors that express B7H3. On Arm A of the study, research participants will receive B7H3-specific CAR T cells only. On Arm B of the study, research participants will receive CAR T cells directed at B7H3 and CD19, a marker on the surface of B lymphocytes, following the hypothesis that CD19+ B cells serving in their normal role as antigen presenting cells to T cells will promote the expansion and persistence of the CAR T cells. Arm A CAR T cells include the protein EGFRt and Arm B CAR T cells include the protein HER2tG. These proteins can be used to both track and destroy the CAR T cells in case of undue toxicity. The primary objectives of the study will be to determine the feasibility of manufacturing the cell products, the safety of the T cell product infusion, to determine the maximum tolerated dose of the CAR T cells products, to describe the full toxicity profile of each product, and determine the persistence of the modified cell in the participant's body on each arm. Participants will receive a single dose of T cells comprised of two different subtypes of T cells (CD4 and CD8 T cells) felt to benefit one another once administered to the research participants for improved potential therapeutic effect. The secondary objectives of this protocol are to study the number of modified cells in the patients and the duration they continue to be at detectable levels. The investigators will also quantitate anti-tumor efficacy on each arm. Participants who experience significant and potentially life-threatening toxicities (other than clinically manageable toxicities related to T cells working, called cytokine release syndrome) will receive infusions of cetuximab (an antibody commercially available that targets EGFRt) or trastuzumab (an antibody commercially available that targets HER2tG) to assess the ability of the EGFRt on the T cells to be an effective suicide mechanism for the elimination of the transferred T cell products.

Inclusion Criteria:

• Participants age ≤ 26 years at the time of consent for study participation; the first 2 participants enrolled and treated with CAR T cells in both Arms A and B will be ≥ 15 years. and ≤ 26 years at time of consent for study participation
• Histologically diagnosed malignant, non-primary CNS solid tumor
• Evidence of refractory or recurrent disease
• Lansky or Karnofsky score ≥ 50
• Life expectancy ≥ 8 weeks
• Recovered from significant acute toxic effects of all prior chemotherapy, immunotherapy and radiotherapy
• If no apheresis product or usable T cell product is available, all chemotherapy has been discontinued ≥ 7 days prior to enrollment
• If no apheresis or usable T cell product is available, all biologic therapy has been discontinued ≥ 7 days prior to enrollment
• If no apheresis product or T cell product is available, all systemic corticosteroid therapy has been discontinued ≥ 7 days prior to enrollment (physiologic replacement dosing is allowed)
• If no apheresis product or usable T cell product is available, at least 3 half-lives or 30 days (whichever is shorter) from time of last dose of anti-tumor directed antibody therapy (including checkpoint inhibitor) at time of enrollment
• If no apheresis product or usable T cell product is available, at least 6 weeks post last dose of myeloablative therapy and autologous and/or allogeneic stem cell transplant, or non-myeloablative therapy and allogeneic stem cell transplant (all timed from stem cell infusion). Participants who receive autologous stem cell infusion following non-myeloablative therapy are eligible once all other eligibility requirements are met.
• If no apheresis product or usable T cell product is available, participants who have received genetically modified cell therapy must be at least 30 days from most recent cell infusion prior to enrollment
• If no apheresis product or usable T cell product is available, participants with neuroblastoma must be at least 12 weeks from I131 MIBG therapy.
• Adequate organ function
• Adequate laboratory values
• Participant is able to tolerate apheresis (including placement of temporary apheresis catheter, if necessary), or already has an apheresis product available for use in manufacturing.
• Participants of childbearing potential must agree to use highly effective contraception

Exclusion Criteria:

• Presence of active malignancy other than primary malignant solid tumor diagnosis
• Current relevant CNS pathology
• Receiving external beam radiation therapy at time of enrollment
• Presence of active GVHD, or receiving immunosuppressive therapy for treatment or prevention of GVHD within 4 weeks prior to enrollment
• Participant is pregnant or breastfeeding
• Participant has presence of active severe infection
• Participant has presence of any condition that, in the option of an investigator, would prohibit the participant from undergoing treatment under this protocol
• Participant has primary immunodeficiency syndrome
• Unwilling or unable to provide consent/assent for participation in the study and 15 year follow up period