pTVG-HP DNA Vaccine With or Without pTVG-AR DNA Vaccine and Pembrolizumab in Patients With Castration-Resistant, Metastatic Prostate Cancer

This trial will evaluate the use of one versus two DNA vaccines, delivered concurrently with PD-1 blockade using pembrolizumab followed by treatment with pembrolizumab alone, and delivered over a prolonged period of time (for a maximum of 2 years (32 cycles) or until radiographic progression) on the treatment of castrate-resistant, metastatic prostate cancer. The hypothesis to be tested is that delivering two vaccines with PD-1 blockade will elicit a greater frequency and magnitude of tumor-directed CD8+ T cells, and thereby increase the percentage of patients experiencing objective anti-tumor effect as measured by PSA declines and/or objective radiographic responses. Participants must be 18 years of age or older and can expect to be on treatment for 2 years (32 cycles) and on study for up to 7 years (including 5 years of follow up via phone).

Inclusion Criteria: * Histologically confirmed diagnosis of prostate cancer (adenocarcinoma of the prostate) * Metastatic disease as evidenced by the presence of soft tissue and/or bone metastases on imaging studies (CT of abdomen/pelvis, bone scintigraphy) * Castrate-resistant disease, defined as follows: * All participants must have received (and be receiving) standard of care androgen deprivation treatment (surgical castration versus GnRH analogue or antagonist treatment); subjects receiving Gonadotropin-releasing hormone (GnRH) analogue or antagonist must continue this treatment throughout the time on this study. * Participants may or may not have been treated previously with a nonsteroidal antiandrogen. For participants previously treated with an antiandrogen, they must be off use of anti-androgen for at least 28 days before day 1 (for flutamide, apalutamide, enzalutamide, or other 2nd generation AR antagonists) or 6 weeks (for bicalutamide or nilutamide) prior to registration. Moreover, participants who demonstrate an anti-androgen withdrawal response, defined as a \> 25% decline in PSA within 4-6 week of stopping a nonsteroidal antiandrogen, are not eligible until the PSA rises above the nadir observed after antiandrogen withdrawal. * Participants must have a castrate serum level of testosterone (\< 50 ng/dL) within 6 weeks of day 1 * Progressive disease while receiving androgen deprivation therapy defined by any one of the following as per the Prostate Cancer Clinical Trials Working Group 3 (PCWG3) bone scan criteria or RECIST 1.1 during or after completing last therapy: * PSA: At least two consecutive rises in serum PSA, obtained at a minimum of 1-week intervals, with the final value \> 2.0 ng/mL. * Measurable disease: \> 50% increase in the sum of the cross products of all measurable lesions or the development of new measurable lesions. The short axis of a target lymph node must be at least 15 mm by spiral CT to be considered a target lesion * Non-measurable (bone) disease: The appearance of two or more new areas of uptake on bone scan (or Sodium Fluoride (NaF) positron emission tomography-computed tomography (PET/CT)) consistent with metastatic disease compared to previous imaging during castration therapy. The increased uptake of pre-existing lesions on bone scan will not be taken to constitute progression, and ambiguous results must be confirmed by other imaging modalities (e.g. X-ray, CT or MRI). * Prior treatment with abiraterone or enzalutamide is permitted, but participants must have weaned to a daily corticosteroid dose equivalent of no more than 5 mg prednisone daily for at least 28 days prior to day 1. * Life expectancy of at least 6 months * Participants must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2. * Adequate hematologic, renal, liver, and coagulation function as evidenced by the following within 6 weeks of day 1: * White Blood Cells (WBC) \>/= 2000 / mm3 * Absolute Neutrophil Count (ANC) \>/= 1500 / mm3 * Hemoglobin (HgB) \>/= 9.0 gm/dL (Participants must not have received a blood transfusion within 14 days) * Platelets \>/= 100,000 / mm3 * Creatinine \