Phase II Trial of Immunotherapeutic HPV Vaccine PRGN-2009 With Pembrolizumab Before Standard Treatment in Subjects With Newly Diagnosed HPV-Associated Oropharyngeal Cancer

* INCLUSION CRITERIA:
* Subjects must have cytologically or histologically confirmed newly diagnosed stage I (T1,2 N1), II or III p16-positive oropharyngeal squamous cell carcinoma (SCC) planned for definitive therapy (surgery or chemoradiotherapy).
* Subjects must have measurable disease, per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1.
* Age \>=18 years.
* Eastern Cooperative Oncology Group \[ECOG\] performance status \<= 2.
* Adequate hematologic function at screening, as follows:

  * Absolute neutrophil count (ANC) \>=1 x 10\^9/L;
  * Hemoglobin (Hgb) \>= 9 g/dL;
  * Platelets \>= 75,000/microliter.
* Adequate renal and hepatic function at screening, as follows:

  * Serum creatinine \<= 1.5 x upper limit of normal (ULN) OR measured or calculated creatinine clearance \>= 40 mL/min for participant with creatinine levels \> 1.5 x ULN (glomerular filtration rate \[GFR\] can also be used in place of creatinine or CrCl);
  * Total bilirubin \<= 1.5 x ULN OR in subjects with Gilbert's syndrome, a total bilirubin \<= 3.0 x ULN;
  * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \<= 2.5 x ULN, unless liver metastases are present, then values must be \<= 3 x ULN.
* Participants serologically positive for human immunodeficiency virus (HIV) Hepatitis B, or Hepatitis C are eligible if the viral loads are undetectable by quantitative polymerase chain reaction (PCR). Note: HIV positive participants must have CD4 count \>= 200 cells/mm\^3 at enrollment, be on stable antiretroviral therapy for at least 4 weeks and have no reported opportunistic infections or Castleman's disease within 12 months prior to enrollment.
* Women of child-bearing potential (WOCBP) must agree to use effective contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation and for at least 4 months following the last dose of pembrolizumab.
* Participants must be willing to undergo two research biopsies on this study.
* Ability of participant to understand and the willingness to sign a written informed consent document.

EXCLUSION CRITERIA:

* Participants with prior investigational drug, live vaccine, chemotherapy, immunotherapy, or any prior radiotherapy (except for palliative bone directed therapy) within the past 4 weeks prior to the first drug administration. Participants may continue adjuvant hormonal therapy in the setting of a definitively treated cancer (e.g., breast).
* Major surgery within 28 days prior to the first drug administration (minimally invasive procedures such as diagnostic biopsies are permitted).
* Pregnant individuals as evaluated by a positive serum or urine Beta-human chorionic gonadotropin (hCG) at screening
* Active autoimmune disease that might deteriorate when receiving an immunostimulatory agent with the exception of:

  * Diabetes type I, eczema, vitiligo, alopecia, psoriasis, hypo- or hyperthyroidism or other mild autoimmune disorders not requiring immunosuppressive treatment.
  * Administration of glucocorticoids through a route known to result in a minimal systemic exposure (topical, intranasal, intraocular, or inhalation).
* Systemic (intravenous or oral) glucocorticoid (except for physiologic doses of corticosteroids, i.e., \<= the equivalent of prednisone 10 mg/day) or other immunosuppressors such as azathioprine or cyclosporin A, are excluded because of potential immune suppression. These treatments must be discontinued at least 1 week prior to enrollment for recent short course use (\<= 14 days). Glucocorticoids as premedication for contrast-enhanced studies is allowed prior to enrollment and on study.
* Participants with a prior or concurrent malignancy whose natural history or treatment that has potential to interfere with the safety or efficacy assessment of the regimen.
* Prior allogenic tissue/solid organ transplant.
* Participants with pulse oximetry \< 92% on room air at screening.
* Uncontrolled intercurrent illness that would limit compliance with study requirements suggested by medical history, physical examination or standard clinical assessments such as imaging and laboratory studies.