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

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 \</= 1.5 x institutional upper limit of normal (ULN) * Total bilirubin \</= 1.5 x institutional ULN OR direct bilirubin \</= ULN for participants with total bilirubin levels \> 1.5 x ULN * Aspartate Aminotransferase (AST), Alanine Aminotransferase (ALT) \</= 2.5 x institutional upper limit of normal * Prothrombin Time (PT) or International Normalized Ratio (INR) \</= 1.5 x ULN unless participant is receiving anticoagulant therapy and PT is within therapeutic range of intended use of anticoagulant (only required for participants receiving biopsy) * Partial Thromboplastin Time (PTT) \</= 1.5 x ULN unless participant is receiving anticoagulant therapy and a PTT is within therapeutic range of intended use of anticoagulant (only required for participants receiving biopsy) * No known history of human immunodeficiency viruses (HIV 1 and 2), Human T-cell leukemia virus type 1 (HTLV-1), or active Hepatitis B or Hepatitis C * Participants must be at least 4 weeks from any prior treatments and have recovered (to \< Grade 2) from acute toxicity attributed to this prior treatment, unless considered chronic * A subset of participants (6 participants per treatment arm) treated at the lead University of Wisconsin (UW) site must be willing and able (in the opinion of the treating physician) to undergo two research biopsies for the investigational component of this trial. * A subset of participants (6 participants per treatment arm) treated at the lead UW site must be willing to undergo NaF PET/CT scans for the investigational component of this trial. * For those participants who are sexually active, they must be willing to use barrier contraceptive methods, and refrain from donating sperm, during the period of treatment on this trial and for four weeks after the last DNA immunization treatment * Participants must be informed of the experimental nature of the study and its potential risks, and must sign an Institutional Review Board (IRB)-approved written informed consent form indicating such an understanding Exclusion Criteria: * Small cell or other variant (non-adenocarcinoma) prostate cancer histology, unless there is evidence that the tumor expresses PAP * Participants may not be receiving other investigational agents or be receiving concurrent anticancer therapy other than standard androgen deprivation therapy * Concurrent bisphosphonate therapy is not excluded, however participants should not start bisphosphonate therapy while on this study; those participants already receiving bisphosphonate therapy should continue at the same dosing and schedule as prior to study entry * Rapidly progressive symptomatic metastatic disease, as defined by the need for increased opioid analgesics within one month of registration for the treatment of pain attributed to a prostate cancer metastatic lesion; participants receiving opioids must receive approval from the PI for eligibility * Treatment with any of the following medications within 28 days of day 1, or while on study, is prohibited: * Systemic corticosteroids (at doses over the equivalent of 5 mg prednisone daily); inhaled, intranasal or topical corticosteroids are acceptable * Prostate Cancer and spes (PC-SPES) * Megestrol * Ketoconazole * 5-α-reductase inhibitors - participants already taking 5-α-reductase inhibitors prior to 28 days prior to registration may stay on these agents throughout the course of therapy, but these should not be started while participants are on study * Diethyl stilbesterol * Abiraterone * Enzalutamide * Apalutamide * Radium 223 (Xofigo®) * Any other hormonal agent or supplement being used with the intent of cancer treatment must be reviewed by the PI for eligibility * External beam radiation therapy within 4 weeks of registration is prohibited, or anticipated need for radiation therapy (e.g. imminent pathological fracture or spinal cord compression) within 3 months of registration. Participants must have recovered from all radiation-related toxicities and not have had radiation pneumonitis. * Major surgery within 4 weeks of registration is prohibited * Prior cytotoxic chemotherapy (for example, but not limited to, docetaxel, mitoxantrone, cabazitaxel) within 28 days of registration is prohibited * Prior treatment with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent, or with any agent directed to another T-cell stimulatory or inhibitory receptor (e.g. CTLA-4, OX-40, CD137). * Participants with a history of life-threatening autoimmune disease * Participants with a history of non-infectious pneumonitis that required corticosteroid treatment, or has current pneumonitis * Participants with a history of allergic reactions to the tetanus vaccine * Participants who have undergone splenectomy or who have a diagnosis of immunodeficiency * Participants must not have other active malignancies other than non-melanoma skin cancers or superficial (non-muscle-invasive) carcinoma of the bladder. Participants with a history of other cancers who have been adequately treated and have been recurrence-free for \> 3 years are eligible. * Participants with known brain metastases and/or carcinomatous meningitis * Participants who have received a live vaccine within 30 days prior to the first dose of study drug. Examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, varicella/zoster (chicken pox), yellow fever, rabies, Bacillus Calmette - Guérin (BCG), and typhoid vaccine. Seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed; however, intranasal influenza vaccines (eg, FluMist®) are live attenuated vaccines and are not allowed. * Any antibiotic therapy within 1 month of day 1, or anticipated need for antibiotic therapy within 1 month of beginning treatment * Participants with active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (eg., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. * Any other medical intervention or condition, which, in the opinion of the PI or treating physician, could compromise participant safety or adherence with the study requirements (including biopsies), or confound results of the study, over the treatment period. * Any known psychiatric or substance abuse disorders that would interfere with cooperation with the requirement of the trial. * Participants cannot have concurrent enrollment on other phase I, II, or III investigational treatment studies.