HPV Vaccine PRGN-2009 Alone or in Combination With Anti-PDL1/TGF-Beta Trap (M7824) in Subjects With HPV Associated Cancers

Background * Metastatic human papillomavirus (HPV) associated malignancies (cervical, anal, oropharyngeal cancers, etc.) are often incurable and poorly palliated by standard therapies. * HPV-positive (p16+) oropharyngeal cancers are the most common HPV-associated malignancy in the United States and are increasing in incidence. * Stage II and III HPV-positive oropharyngeal cancer is primarily treated with definitive therapy. * Although the prognosis for stage I HPV+ oropharyngeal cancer is favorable, about 20 percent of patients with stage II disease and 35 percent of patients with stage III disease will die within four years. * Attempts to de-intensify treatment of HPV-positive oropharyngeal cancer by replacing high-dose cisplatin with cetuximab concurrent with radiotherapy have failed. * Induction and neoadjuvant immunotherapy are an area of active study in this type of cancer. The aims of induction immunotherapy are to induce antigen-specific immunity prior to definitive therapy and to reduce the risk of disease relapse for patients with stage II and III disease. * Therapeutic vaccines targeting HPV alone or in combination with M7824 (MSB0011359C) (dual programmed death-ligand 1 (PD-L1) and transforming growth factor beta (TGF- beta) inhibitor) have demonstrated induction of HPV antigen-specific responses and tumor growth inhibition in multiple pre-clinical models of HPV-positive malignancy. * In clinical studies done in the Center for Cancer Research (CCR), M7824 as monotherapy has produced a notable objective response rate (35-40%) for metastatic HPV + cancers including Oropharyngeal Squamous Cell Carcinoma (OPSCC) and preclinical studies support the addition of an investigational HPV vaccine with therapeutic intent (PRGN-2009, a gorilla adenoviral based vaccine) to further increase anti-tumor efficacy. Objectives: Phase I in participants with recurrent/metastatic HPV positive cancer: -Primary objective: To determine the safety and recommended phase II dose (RP2D) of PRGN-2009 (HPV vaccine) alone or in combination with M7824 administered at RP2D of 1200 mg every 2 weeks (q2w). Phase II in participants with newly diagnosed stage I (T1, T2 N1)/II/III p16-positive oropharyngeal cancer and patients with newly diagnosed operable stage II/III/IVA/IVB/HPV + sinonasal squamous cell cancer: -Primary objective: To determine if HPV vaccine alone (Arm 2A) is able to result in a \>= 2-fold increase in cluster of differentiation 3 (CD3+) tumor infiltrating T cells post treatment compared with pre-treatment in p16-positive oropharyngeal cancer. Eligibility: Phase I: * Men or women of age \>= 18 years old. * Subjects with cytologically or histologically confirmed locally advanced not amenable to potentially curative local therapies or metastatic HPV associated malignancies: * Cervical cancers; * p16+ Oropharyngeal cancers; * Anal cancers; * Vulvar, vaginal, penile, and squamous cell rectal cancers * Other locally advanced or metastatic solid tumors (e.g., lung, esophagus) that are known HPV+. * Prior first line systemic therapy is required Phase II: * Men or women of age \>= 18 years old. * Subjects with newly diagnosed stage I (T1, T2 N1), II or III, II or III p16-positive oropharyngeal squamous cell carcinoma (OPSCC) or stage II/III/IVA/IVB HPV-SNSCC planned for definitive therapy. Design: Phase I: Recurrent/metastatic HPV associated cancer: * A 3+3 dose escalation design will be used which will evaluate PRGN-2009 (HPV vaccine) at two dose levels (1x10\^11 and 5x10\^11 viral particle (VP) units) given as monotherapy followed by a third dose level evaluating the RP2D dose of PRGN-2009 in combination with 1200 mg (RP2D) of M7824. In addition, the combination of PRGN-2009 at RP2D with 1200 mg of M7824 will be expanded to a total of 10 evaluable participants to gauge the preliminary efficacy of the combination of PRGN-2009 and M7824 in participants with advanced disease. * There will be a 4-week dose limiting toxicity (DLT) evaluation period for each dose level. * It is expected that up to 22 participants may enroll. Phase II: Newly diagnosed p16-positive oropharyngeal cancer: * Evaluation of HPV vaccine alone (Arm 2A: Stage I (T1,T2 N1)/II/III) as neoadjuvant/ induction therapy before definitive standard of care therapy. * Participants will receive neoadjuvant/ induction immunotherapy at National Institutes of Health (NIH) Clinical Center and then be referred back to their home institution for definitive standard of care therapy. * It is expected that up to 20 participants may enroll. * Newly diagnosed stage II/III/IVA/IVB HPV-SNSCC: * Enrollment and treatment will occur similarly as participants with p16+oropharyngeal cancer for exploratory correlates to advise possible future trials. Up to 2 participants may enroll in this group.