Vaccine Therapy Plus Pembrolizumab in Treating Advanced Ovarian, Fallopian Tube, or Primary Peritoneal Cavity Cancer

Inclusion Criteria:

* Age \>= 18 years
* Histologically confirmed recurrent epithelial ovarian, fallopian tube, or primary peritoneal cancer NOTE: Histologic confirmation of the primary tumor or recurrent tumor per pathology report is required. Eligible histotypes include high grade serous; endometrioid; and clear cell carcinoma, as these histotypes have high expression of FRalpha. Mixed carcinomas, including carcinosarcomas, with \>= 50% of the tumor comprised of high grade serous; and/or endometrioid; and/or clear cell carcinoma are eligible
* Ovarian cancer (OC) recurrence - Platinum sensitivity/resistance

  * Platinum-refractory (defined as recurrence or progression of OC =\< 30 days of the last dose of platinum-based chemotherapy)
  * Platinum-resistant (defined as recurrence or progression of OC between 31-180 days of the last dose of platinum-based chemotherapy)
  * Platinum-sensitive (defined as recurrence or progression \>=181 days after the last dose of platinum-based chemotherapy).

NOTE: Any number of prior therapies or maintenance regimens for OC are allowed

* At least one of the following:

  * Measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 criteria AND/OR
  * CA-125-evaluable disease, as defined by the Gynecologic Cancer InterGroup (GCIG)
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1
* Hemoglobin \>= 8.5 g/dL (obtained =\< 15 days prior to registration)
* Absolute neutrophil count (ANC) \>= 1000/mm\^3 (obtained =\< 15 days prior to registration)
* Platelet count \>= 75,000/mm\^3 (obtained =\< 15 days prior to registration)
* Lymphocytes \>= 0.3 x 10\^9/L (obtained =\< 15 days prior to registration)
* Monocytes \>= 0.25 x 10\^9/L (obtained =\< 15 days prior to registration)
* Total bilirubin =\< 1.5 x upper limit of normal (ULN), unless patient has a documented history of Gilbert's disease, then direct bilirubin must be =\< ULN (obtained =\< 15 days prior to registration)
* Aspartate transaminase (AST) =\< 3 x ULN (obtained =\< 15 days prior to registration)
* Creatinine clearance \>= 30 mL/min per Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) creatinine equation (obtained =\< 15 days prior to registration)
* Negative pregnancy test done =\< 7 days prior to registration, for persons of childbearing potential only
* Provide written informed consent
* Willing to provide mandatory blood and tissue specimens for correlative research
* Willing to provide archival tissue specimen for correlative research
* Willing to return to Mayo Clinic for follow-up (during the active monitoring phase of the study)
* Willing to undergo a tetanus vaccination (if not performed =\< 365 days prior to registration)
* Willing to have a temporary central access line placed for apheresis, if needed

Exclusion Criteria:

* Any of the following because this study involves an investigational agent, the genotoxic, mutagenic and teratogenic effects of which on the developing fetus and newborn are unknown

  * Pregnant persons
  * Nursing persons
  * Persons of childbearing potential or able to father a child who are unwilling to employ adequate contraception
* Prior treatment for ovarian cancer with an anti-PD-1 or anti-PD-L1 monoclonal antibody
* Treatment with IV anti-cancer therapy =\< 3 weeks prior to registration or with oral anti-cancer therapy =\< 1 week prior to registration NOTE: Since treatment will begin no sooner than 4 weeks after registration due to the need for apheresis and manufacturing of the FRαDC product, a "wash-out" period prior to registration will cause a gap of at least 5 weeks between the last anti-cancer treatment and initiation of protocol therapy
* Grade 2 or higher symptoms attributed to OC OR disease measuring \> 5 cm in long axis (non-nodal lesions), or \> 5 cm in short axis (nodal lesions) OR disease that, in the judgement of the treating investigator, is likely to become symptomatic in the next 8 weeks (ex. moderate ascites)

  * NOTE: Since patients will not receive therapy for cancer until 3-4 weeks after apheresis--which is potentially 6-8 weeks after registration --patients with symptomatic OC or an elevated tumor burden may experience significant progression prior starting therapy and should not be treated on this protocol).
* Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens
* Uncontrolled human immunodeficiency virus (HIV) infection and/or HIV-infected patients with a history of Kaposi's sarcoma and/or multicentric Castleman disease.

  * NOTE: HIV-infected participants must have well-controlled HIV on anti-retroviral therapy (ART), defined as:

    * Participants on ART must have a CD4+ T-cell count ≥ 350 cells/mm\^3 at the time of screening
    * Participants on ART must have achieved and maintained virologic suppression defined as confirmed HIV ribonucleic acid (RNA) level below 50 or the lower limit of quantification derivation technique (LLOQ) (below the limit of detection) using the locally available assay at the time of screening and for at least 12 weeks before screening
    * It is advised that participants must not have had any AIDS-defining opportunistic infections within the past 12 months
    * Participants on ART must have been on a stable regimen, without changes in drugs or dose modification, for at least 4 weeks before study entry (day 1) and agree to continue ART throughout the study
  * NOTE: No HIV testing is required unless mandated by local health authority
* Uncontrolled intercurrent illness including, but not limited to:

  * Ongoing or active serious infections (e.g., pneumonia, sepsis) requiring systemic therapy
  * Current diagnosis or previous history of immune-related (non-infectious) pneumonitis or interstitial lung disease that requires or required steroids
  * Active autoimmune disease that required systemic treatment other than replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroids) =\< 2 years prior to registration
  * Psychiatric illness/social situations that would limit compliance with study requirements
  * Concurrent active hepatitis B \[defined as hepatitis B surface antigen (HBsAg) positive and/or detectable hepatitis B virus (HBV) deoxyribonucleic acid (DNA) \] and Hepatitis C virus \[defined as anti-hepatitis C virus (HCV) antibody (Ab) positive and detectable HCV RNA\] infection. EXCEPTIONS:

    * For patients with evidence of hepatitis B virus (HBV) infection (HBsAg positive), patients must have completed at least 4 weeks of hepatitis B virus (HBV) antiviral therapy and the HBV viral load must be undetectable at the time of registration

      * NOTE: Patients should remain on anti-viral therapy throughout study intervention and follow local guidelines for HBV anti-viral therapy post completion of study intervention
    * Patients with a history of hepatitis C virus (HCV) are eligible if they have an undetectable HCV viral load.

      * NOTE: Patients must have completed curative anti-viral treatment \>= 4 weeks prior to registration
  * NOTE: Patients without symptoms or prior history do not require testing prior to registration unless mandated by local health authority
* Other active malignancy either requiring palliative systemic therapy =\< 3 years prior to registration, or likely to require treatment in the next 2 years EXCEPTIONS: Patients with non-melanotic skin cancer, papillary thyroid cancer not requiring therapy or carcinoma-in-situ are eligible for this trial. Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial
* History of myocardial infarction =\< 6 months prior to registration, or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias NOTE: Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better
* Treatment with systemic immunosuppressive medication (including, but not limited to, prednisone \>10 mg/day or equivalent, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor \[TNF\]-alpha agents) =\< 7 days prior to registration, or anticipation of need for systemic immunosuppressive medication during the course of the study NOTE: Patients who have received acute, low-dose systemic steroids (=\< 10 mg/day oral prednisone or equivalent) prior to registration or a one-time pulse dose of systemic immunosuppressant medication (e.g., =\< 48 hours of corticosteroids for a contrast allergy) are eligible for the study NOTE: The use of inhaled corticosteroids for chronic obstructive pulmonary disease or asthma, mineralocorticoids (e.g., fludrocortisone), or low-dose corticosteroids for patients with orthostatic hypotension or adrenocortical insufficiency is allowed
* History of allogeneic stem cell transplant