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

* 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 months) or clinically significant cerebrovascular accident (\<3 months). In order to be eligible participants must have repeated central nervous system (CNS) imaging at least a month after definitive treatment showing stable CNS disease. Participants with evidence of intra-tumoral or peritumoral hemorrhage on baseline imaging are also excluded unless the hemorrhage is grade =\< 1 and has been shown to be stable on two consecutive imaging scans.
* Pregnant women are excluded from this study because M7824 and PRGN-2009 vaccine have not been tested in pregnant women and there is potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with these immunotherapies, breastfeeding should be discontinued if the mother is treated on this protocol.
* Only for Phase I, Arm 1B: Active autoimmune disease that might deteriorate when receiving an immunostimulatory agent with exception of:

  * Diabetes type I, eczema, vitiligo, alopecia, psoriasis, hypo- or hyperthyroid disease or other mild autoimmune disorders not requiring immunosuppressive treatment;
  * Administration of steroids for other conditions through a route known to result in a minimal systemic exposure (topical, intranasal, intro-ocular, or inhalation) is acceptable;
  * Subjects on systemic intravenous or oral corticosteroid therapy with the exception of

physiologic doses of corticosteroids (=\< the equivalent of prednisone 10 mg/day) or other immunosuppressors such as azathioprine or cyclosporin A are excluded on the basis of potential immune suppression. For these subjects these excluded treatments must be discontinued at least 1 weeks prior to enrollment for recent short course use (=\< 14 days) or discontinued at least 4 weeks prior to enrollment for long term use (\> 14 days). In addition, the use of corticosteroids as premedication for contrast-enhanced studies is allowed prior to enrollment and on study.

* Only for Phase I: Subjects with a history of serious intercurrent chronic or acute illness, such as cardiac or pulmonary disease, hepatic disease, bleeding diathesis or recent (within 3 months) clinically significant bleeding events, known left ventricular ejection fraction \<50% (confirmation of ejection fraction (EF) \> 50% is not required for eligibility), history of myocarditis, or recent myocardial infarction (within 6 months), or other illness considered by the Investigator as high risk for M7824 drug treatment.
* Only for Phase I: Subjects refusing to accept blood products as medically indicated.
* History of second malignancy within 3 years of enrollment except for the following: adequately treated localized skin cancer, cervical carcinoma in situ, superficial bladder cancer, other localized malignancy which has been adequately treated or malignancy which does not require active systemic treatment (e.g., low risk chronic lymphocytic leukemia (CLL)). For participants enrolled on the phase I portion of the protocol a second HPV driven malignancy is allowed.
* Only for Phase I, Arm 1B: Subjects with a known severe hypersensitivity reaction to monoclonal antibodies or its excipients (grade \>/= 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).