FGFR4 Chimeric Antigen Receptor (CAR) T Cells in Children and Young Adults With Recurrent or Refractory Rhabdomyosarcoma

* INCLUSION CRITERIA * Histologically confirmed rhabdomyosarcoma by the NCI Department of Pathology. Note: Since FGFR4 expression is universal in rhabdomyosarcoma, confirmation of FGFR4 expression is not required. * Relapsed or refractory rhabdomyosarcoma after at least two (2) cancer treatment regimens i.e., participants should have relapsed or progressed after upfront therapy (that includes any systemic chemotherapy with or without local control) as well as at least one salvage therapy (which can be systemic therapy, radiation, or surgery). * No available alternative curative therapies per standard of care. * Participants must have measurable disease per RECIST 1.1 or non-measurable disease on imaging. * Age \>= 3 and \<= 39 years old. * Weight \>=15 kg. * Performance status: Karnofsky \>= 50% (\>= 16 years) or Lansky \>= 50% (\< 16 years). Note: Participants who are unable to walk because of paralysis, but who are upright in a wheelchair will be considered ambulatory for calculating the performance score. * Participants must be willing to accept blood transfusions. * Adequate organ and marrow function as defined below: * Organ: Bone Marrow Function\* * Laboratory Element: Absolute neutrophil count; Minimum Requirement \>= 500/mcL * Laboratory Element: Platelets; Minimum Requirement \>= 50,000/mcL \*Transfusion independent (defined as no transfusion in the prior 7 days) for participants without bone marrow involvement. Participants who have bone marrow involvement with tumor are exempt from the platelet requirement and will not be evaluable for hematological toxicities. Participants must not be refractory to transfusions. * Organ: Liver Function * Laboratory Element: Aspartate aminotransferase (AST); Minimum Requirement \<= 5 x upper limit of normal (ULN) * Laboratory Element: Alanine aminotransferase (ALT); Minimum Requirement \<= 5 x ULN * Laboratory Element: Total bilirubin; Minimum Requirement \<= 2 x ULN (Note: Participants with Gilbert's syndrome and/or bilirubin elevation due to tumor involvement are allowed to have \<= 5 x ULN) Note: Adult values will be used for calculating hepatic toxicity and determining eligibility --Organ: Renal Function * Age: 3 to \< 6 years; Maximum serum creatinine (mg/dL): Male - 0.8, Female - 0.8 * Age: 6 to \< 10 years; Maximum serum creatinine (mg/dL): Male - 1, Female - 1 * Age: 10 to \< 13 years; Maximum serum creatinine (mg/dL): Male - 1.2, Female - 1.2 * Age: 13 to \< 16 years; Maximum serum creatinine (mg/dL): Male - 1.5, Female - 1.2 * Age: \>= 16 years; Maximum serum creatinine (mg/dL): Male - 1.7, Female - 1.4 OR * Measured or calculated creatinine clearance or glomerular filtration rate (GFR); Minimum Requirement: \>= 60mL/min/1.73 m\^2 --Organ: Cardiac Function * Laboratory Element: Cardiac status; Minimum Requirement: Cardiac ejection fraction \>= 45% or shortening fraction \>= 28%, pericardial effusion \<= grade 2 as determined by an echocardiogram (ECHO) * Organ: Pulmonary Function * Laboratory Element: Pulmonary status; Minimum Requirement: Pleural effusion \<= grade 1; Oxygen (O2) saturation \>=92% on room air at rest --Organ: Neurological Function * Laboratory Element: Neurologic status; Minimum Requirement: No acute neurotoxicity greater than grade 2 per CTCAE v.5.0 with the exception of decreased tendon reflex (DTR). Any grade of DTR is eligible. * Individuals of child-bearing potential (IOCBP) must agree to use highly effective contraception (hormonal, intrauterine device \[IUD\], abstinence, surgical sterilization) at the study entry and up to 12 months after the last dose of combined chemotherapy or 6 months after FGFR4-CAR T cells infusion, whichever is later. Individuals who can father children must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) at the study entry and up to 4 months after the last dose of combined chemotherapy or 6 months after FGFR4-CAR T cells infusion, whichever comes later. We also will recommend individuals who can father children with IOBCP partners ask their partners to be on highly effective birth control (hormonal, IUD, surgical sterilization). Individuals who can father children must not freeze or donate sperm within the same period. * Nursing participants must be willing to discontinue nursing from study treatment initiation through 4 months after completion of chemotherapy preparative administration or 6 months after FGFR4-CAR T cells infusion, whichever is later. * Participants with previous central nervous system (CNS) tumor involvement that has been treated and is stable for at least 6 weeks following completion of therapy as evidenced by no requirements for corticosteroids, no evolving neurologic deficits, and no progression of residual brain abnormalities without specific therapy, are permitted. Participants with asymptomatic subcentemeric CNS lesions are permitted if no immediate radiation or surgery is indicated. * Participants must be willing to be enrolled into protocol 15C0028 "Follow-Up Evaluation for Gene-Therapy Related Delayed Adverse Events after Participation in Pediatric Oncology Branch Clinical Trials" after 5 years on this trial. * The ability of participant or parent/guardian to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA * Prior therapy with the following prior to apheresis: * tyrosine kinase inhibitor, targeted agent, or metronomic non-myelosuppressive regimen within \<= 1 week * systemic chemotherapy within \<= 2 weeks * antineoplastic antibody therapy, checkpoint inhibitors, or vaccine therapy, within \<= 3 weeks or 5 half-lives (whichever is shorter) * radiation within \<= 3 weeks (\<= 6 weeks if CNS or lung fields have been radiated or in case of craniospinal irradiation of radiation of \>=50% of bony pelvis and \<=12 weeks in case of total body irradiation). Note: There is no time restriction if the volume of bone marrow treated is less than 10% and the participant has measurable/evaluable disease outside the radiation port * any investigational agents within \<= 4 weeks * autologous stem cell infusion following myeloablative therapy within \<= 6 weeks * genetically modified T cell, NK cell, or dendritic cell therapy within \<= 6 weeks * allogeneic stem cell transplant/infusion within \<=12 weeks or evidence of active graft versus host disease (GVHD) * Participants receiving more than physiologic dosing of systemic steroids (3 mg/m\^2/day of prednisone equivalent). * History of severe, immediate hypersensitivity reaction attributed to any agents used in the study or in the manufacturing of the cells. * Second malignancy at any time. * Primary immunodeficiency. * Seropositive for human immunodeficiency virus (HIV) antibody. * Seropositive for hepatitis C (HCV) or positive for Hepatitis B (HBV) surface antigen (HbsAg). * Pregnancy confirmed with beta-HCG serum or urine pregnancy test performed in IOCBP at screening. * Uncontrolled intercurrent illness or social situations that would limit compliance with study requirements.