Phase II Trial of Neoadjuvant Chemotherapy (NAC) Alone or in Combination With Immunotherapy Vaccine PRGN-2009 in Subjects With Newly Diagnosed HPV-Associated Oropharyngeal (Head and Neck) Cancer

* INCLUSION CRITERIA: * Histologically or cytologically confirmed surgically resectable newly diagnosed stage I (cT1-2, N0-1) or II (T1-3, N0-2), M0 oropharyngeal squamous cell carcinoma. Note: Pathological report of cancer diagnosis may be from the primary tumor or from a metastatic cervical lymph node. * History of HPV-positive status determined by a standard-of-care HPV testing. Note: All participants with high-risk HPV serotypes are eligible. * Age \>= 18 years. * ECOG performance status \<= 2. * Individuals who smoke currently must smoke \<10 pack years. Note: Former smokers with any pack-year history are eligible if quit smoking \>10 years before study treatment initiation. Former smokers who quit \<10 years before study treatment initiation must have smoked \<10 pack years. * Planned for cancer removal surgery per standard of care (SOC) and individual had agreed for the cancer removal surgery. * Individuals must have adequate organ and marrow function as defined below: * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L * Hemoglobin (Hgb) \>= 9.0 g/dL * Platelet count \>= 100 x 10\^9/L * Creatinine \<= 1.2 x upper limit of normal (ULN) OR calculated creatinine clearance \>=55 mL/min/1.73m\^2 by Cockcroft-Gault formula * Total bilirubin \<= 1 x ULN, or \<= 3 x ULN in patients with known or suspected Gilbert's Syndrome * Alanine aminotransferase (ALT) \<= 1.5 x ULN * Aspartate aminotransferase (AST) \<= 1.5 x ULN * Individuals serologically positive for human immunodeficiency virus (HIV) must: * be on effective anti-retroviral therapy for at least 4 weeks; and * have undetectable viral load; and * have the CD4 count \>=200 cells/microL; and * have no reported opportunistic infections or Castleman s disease within 12 months prior to study treatment initiation * Individuals serologically positive for Hepatitis C virus (HCV) or Hepatitis B virus (HCB) must have an undetectable viral load. * Individuals of child-bearing potential (IOCBP) must agree to use a highly effective method of contraception (hormonal, intrauterine device (IUD), surgical sterilization, abstinence) for the duration of the study treatment and up to 2 months after the last dose of PRGN-2009 and an effective method of contraception (barrier, hormonal, intrauterine device (IUD), surgical sterilization, abstinence) for 14 months after the last dose of cisplatin/docetaxel. Note: IOCBP is defined as any woman who has experienced menarche and has not had a hysterectomy or bilateral oophorectomy or is not postmenopausal (amenorrheic 12 months or more following cessation of exogenous hormonal treatments; if \<50 years old and need follicle stimulating hormone \[FSH\] in the post-menopausal range). Men must agree to use a highly effective method of contraception (surgical sterilization, abstinence) for the duration of the study treatment and up to 2 months after the last dose of PRGN-2009 and an effective method of contraception (barrier, surgical sterilization, abstinence) for the duration of the study treatment and up to 11 months after the last dose of the study drug(s). We also will recommend men on treatment with PRGN-2009 with female partners of childbearing potential ask female partners to be on highly effective birth control (hormonal, intrauterine device (IUD), surgical sterilization) during PRGN-2009 treatment and 2 months after that. * Breastfeeding individuals must be willing to discontinue breastfeeding from study treatment initiation through 2 months after the last dose of the study drug(s). * Individuals must have a tumor site that is amenable to biopsy and be willing to undergo pre- treatment biopsy for research purposes. * Individuals must be willing to undergo pre-treatment PET/CT imaging study. * The ability of an individual to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: * Peripheral motor or sensory neuropathy \> Grade 2 per Common Terminology Criteria for Adverse Events (CTCAE) v.5 at screening. * Prior therapy with an investigational drug, live vaccine, chemotherapy, immunotherapy, or any prior radiotherapy (except for palliative bone-directed therapy) within 4 weeks prior to the first study drug administration. Note: Participants may continue adjuvant hormonal therapy in the setting of a definitively treated cancer (e.g., breast). * Prior therapy with any medications or substances that are moderate or strong inducers or moderate or strong inhibitors of cytochrome P450 (CYP3A) https://www.fda.gov/Drugs/DevelopmentApprovalProcess/DevelopmentResources/DrugInteractionsLabeling/ucm093664.htm#table2-2,table3-3,table5-2 within 2 weeks prior to the first study drug administration. * History of allergic reactions attributed to compounds of similar chemical or biological composition to drugs used in the study. * 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 within 1 week prior to study treatment initiation. Note: Glucocorticoids as premedication for contrast-enhanced studies are allowed. * Second malignancy active within the previous 2 years except for indolent or locally curable malignancy that is currently considered cured and/or does not require an additional standard of care treatment, such as, but not limited to, cutaneous basal or squamous cell carcinoma, superficial bladder cancer, or cervical carcinoma in situ, or an incidental histological finding of prostate cancer or differentiated thyroid cancer. * Prior allogenic tissue/solid organ transplant. * History of heart failure. * Positive beta-human chorionic gonadotropin (beta-HCG) serum or urine pregnancy test performed in females of childbearing potential at screening. * Uncontrolled intercurrent illness or medical condition(s) evaluated by medical history and physical exam or situations that are not stable (e.g., recent hospitalization, Emergency Room visit or undergoing medication changes) that would potentially increase risk for the participant.