Modified Immune Cells (Autologous CAR T Cells) in Treating Patients with Advanced, Recurrent Platinum Resistant Ovarian, Fallopian Tube or Primary Peritoneal Cancer

Inclusion Criteria:

* Women with recurrent, advanced, platinum resistant ovarian, fallopian tube, and primary peritoneal cancer that have progressed after receiving standard of care therapies or are not eligible to receive available therapies with known clinical benefit will be eligible for the study. Patients must have measurable disease that can be accurately measured by RECIST 1.1 criteria in at least one dimension as \>= 1.0 cm or \> 1.5 cm lymph node with computed tomography (CT), ultrasound, or magnetic resonance imaging (MRI) techniques.

  * Platinum resistant is defined as progression of disease within six months of platinum regimen.
  * Patients with BRCA mutations who have completed standard therapies (including PARP inhibitors) are allowed on this study.
* Patients must be capable of understanding and providing a written informed consent.
* Patients must be 14 days from previous cytotoxic chemotherapy at time of cell collection.
* Laboratory values must indicate adequate organ function.
* Patients must be at least 28 days post systemic steroids prior to enrollment except as premedication for contrast allergy and/or other protocol-mandated medication.
* Patients must have Eastern Cooperative Oncology Group (ECOG) performance status score of =\< 2.
* Patients must have recovered from major acute infections and/or recent surgical procedures, and in the opinion of the investigator, not have any significant active concurrent medical illnesses precluding protocol treatment.
* Negative pregnancy test for women of childbearing potential. Women of childbearing potential are those who have not been surgically sterilized, are \< 60 years old, or have had menses within the past 12 months.
* Women of childbearing potential must be willing to use 2 methods of contraception before, during, and at least 4 months after the PRGN-3005 cell infusion.

Exclusion Criteria:

* Patients with any of the following cardiac conditions:

  * Symptomatic restrictive cardiomyopathy
  * Unstable angina or symptomatic coronary artery disease within 4 months prior to enrollment
  * New York Heart Association functional class III-IV heart failure on active treatment
  * Symptomatic pericardial effusion
  * Congestive heart failure
  * Clinically significant hypotension.
* Patients with CA 125 =\< ULN during screening.
* Patients with history of human immunodeficiency virus (HIV), West Nile, Zika, or active hepatitis B or C infections.
* Patients with severe, symptomatic ascites requiring diuretics, regular paracentesis, or other invasive interventions.
* Patients within 28 days of receiving another investigational agent.
* Patients with pulmonary hypertension, pulmonary fibrosis, or restrictive lung disease, patients with baseline oxygen saturation on room air \< 92%, forced expiratory volume in 1 second (FEV1) =\< 50%, or diffusion capacity of the lung for carbon monoxide (DLco) (corrected) of \< 40% will be excluded.
* Women who are pregnant or breast feeding.
* Patients with second malignancy within the last 5 years excluding basal carcinoma of the skin, squamous carcinoma of the skin, or in situ cervical dysplasia that has undergone curative therapy.
* Patients with an active autoimmune disease requiring immunosuppressive therapy or uncontrolled with treatment.
* Patients who are simultaneously enrolled in any other treatment study.
* Clinical or radiological evidence of acute bowel obstruction within 30 days of signing consent.
* Patients with known or treated brain metastases.
* Patients with an active seizure disorder.
* Any female patient \<60 years old who does not meet at least one of the following criteria will be considered to have reproductive potential:

  * Post-menopausal for at least 12 consecutive months (i.e., no menses), or
  * Undergone a sterilization procedure (hysterectomy, salpingectomy, or bilateral oophorectomy; tubal ligation is not considered a sterilization procedure). Pregnancy test for females of reproductive potential must be negative within 14 days before leukapheresis.