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

Background: For some cancers associated with human papillomavirus (HPV), standard treatments are not helpful. Researchers want to see if a vaccine for HPV combined with a drug called M7824 (MSB0011359C) has a better effect on these cancers than when they work alone. Objective: To find a safe dose of HPV vaccine alone or combined with M7824. Also, to test if either HPV vaccine alone or combined with M7824 causes a better immune response. Eligibility: People ages 18 and older with locally advanced or metastatic HPV associated cancer (Phase I) or stage II or III p16-positive oropharyngeal cancer (Phase II) Design: Participants will be screened with: Medical history Physical exam Blood, urine, and heart tests Possible photos of skin lesions Computed tomography (CT), magnetic resonance imaging (MRI), or nuclear bone scan: Participants will lie in a machine that takes pictures of the body. For the CT scan, they may have a contrast agent injected into a vein. Participants may have up to 2 tumor biopsies. For participants in Phase II, this may be performed with a thin tube placed through the nose into the airway. Participants will receive the HPV vaccine alone or with M7824. For participants on the Phase II, they will receive two doses of HPV vaccine under the skin either alone or with M7824 as an infusion spaced two weeks apart. This will be done prior to their planned chemoradiation or surgery. For participants on the Phase I, they will get the HPV vaccine injected under the skin 2 to 3 times in the first month. Then they will have a booster every 4 weeks. They will receive M7824 as an infusion into a vein every 2 weeks. Treatment will last up to 1 year. After they stop treatment, participants will have a visit within 4 weeks. They will then be contacted for long-term follow-up every year, for the rest of their lives. ...

* INCLUSION CRITERIA: * Subjects with cytologically or histologically confirmed locally advanced not amenable to potentially curative local therapies or metastatic human papillomavirus (HPV) associated malignancies (Phase I only): * 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+. * Subjects with cytologically or histologically confirmed newly diagnosed stage II or III p16-positive oropharyngeal squamous cell carcinoma planned for definitive therapy or with newly diagnosed stage II or III or IVA or IVB HPV-positive sinonasal squamous carcinoma (HPV-SNSCC) eligible for primary surgery (Phase II only). * Subjects must have measurable disease, per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 (Phase 1 only). * Phase I only: Participants must have received one prior line of systemic chemotherapy in the recurrent/metastatic setting as well as checkpoint blockade therapy in tumors with Food and Drug Administration (FDA) approval (head and neck squamous cell cancer and programmed death-ligand 1 (PDL1+) cervical cancer). Exceptions to this include participants not eligible to receive standard therapy. * Men or Women; Age \>=18 years. * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Adequate hematologic function at screening, as follows: * Absolute neutrophil count (ANC) \>=1 x 109/L; * Hemoglobin \>= 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 institutional ULN (GFR can also be used in place of creatinine or CrCl); * 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). * The effects of the immunotherapies (PRGN-2009 vaccine and M7824) on the developing human fetus are unknown. For this reason and because M7824 and PRGN-2009 used in this trial are possibly teratogenic, women of child-bearing potential and men must agree to use highly effective contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and up to 2 months following the last dose of M7824 study treatment. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. * Participants serologically positive for human immunodeficiency virus (HIV), Hep B, Hep C are eligible as long as the viral loads are undetectable by quantitative polymerase chain reaction (PCR). HIV positive participants must have cluster of differentiation 4 (CD4) count \>= 200 cells per cubic millimeter 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. EXCLUSION CRITERIA: * Participants with prior investigational drug, live vaccine, chemotherapy, immunotherapy or any prior radiotherapy (except for palliative bone directed therapy) within the past 28 days prior to the first drug administration except if the investigator has assessed that all residual treatment-related toxicities have resolved or are minimal and feel the participant is otherwise suitable for enrollment. 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). * Known active brain or central nervous system metastasis (less than a month out from definitive radiotherapy or surgery), seizures requiring anticonvulsant treatment (\/= 3 National Cancer Institute (NCI)-Common Terminology Criteria for Adverse Events (CTCAE) v5) will be evaluated by the allergy/immunology team prior to enrollment. * Prior allogenic tissue/solid organ transplant. * For participants who may receive M7824: previous life-threatening side effects resulting from prior checkpoint inhibitor therapy. * Participants with pulse oximetry \< 92% on room air at screening. * Participants unable to provide informed consent. * Participants whose inclusion in the trial would in the judgement of the principal investigator (PI) lead to time from diagnosis to initiation of curative treatment of\>70 days (Arm 2A only).