Vaccine treatment for advanced HPV-related throat cancer

Inclusion Criteria:

* Age \>= 18 years
* Disease characteristics

  * Locally advanced HPV-OPSCC and high-risk HPV-specific testing with at least one of the following:

    * Radiology extranodal extension (ENE) OR
    * cN2 (AJCC 8th Edition) disease (contralateral/bilateral nodes) OR
    * cN3(AJCC 8th Edition) disease (lymph node \[LN\] \> 6 cm) OR
    * Radiographic evidence of 2 or more involved lymph nodes
  * Candidate for curative intent surgery or chemo-radiation
* Measurable or unmeasurable disease as defined by RECIST 1.1 criteria
* Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1
* White blood cell (WBC) count \>= 3,000/mm\^3 (=\< 15 days prior to registration)
* Platelet count \>= 75,000/mm\^3 (=\< 15 days prior to registration)
* Hemoglobin \>= 9.0 g/dL (5.6 mmol/L) (=\< 15 days prior to registration)

  * NOTE: Transfusions are not allowed =\< 7 days prior to registration
* Total bilirubin =\< 1.5 X upper limit of normal (ULN) (or total bilirubin =\< 3.0 X ULN with direct bilirubin =\<1.5 X ULN in patients with well-documented Gilbert's Syndrome) (=\< 15 days prior to registration)
* Aspartate aminotransferase/serum glutamic-oxaloacetic transaminase (AST/SGOT) =\< 2.5 X ULN (=\< 15 days prior to registration)
* Creatinine =\< 1.5 mg/dL (133 umol/L) OR calculated creatinine clearance \>= 30 mL/min/1.73m\^2 for patients with creatinine levels above ULN (=\< 15 days prior to registration)
* Prothrombin time (PT)/international normalized ratio (INR)/partial thromboplastin time (PTT) =\< 1.5 X ULN OR if patient is receiving anticoagulant therapy and PT or PTT is within therapeutic range of intended use of anticoagulants (=\< 15 days prior to registration)
* Negative pregnancy test done =\< 3 days prior to registration for persons of childbearing potential only
* Persons of childbearing potential or able to father a child must be willing to use an effective method of contraception for the course of the study starting with the first dose of study therapy through 120 days after the last dose of study medication

  * NOTE: Abstinence is acceptable if this is the usual lifestyle and preferred method of contraception for the patient
* Provide written informed consent
* Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study)
* Willingness to provide mandatory blood specimens for correlative research
* Willingness to provide mandatory tissue specimens for correlative research

Exclusion Criteria:

* Active autoimmune disease requiring systemic treatment, documented history of severe autoimmune disease, or a syndrome that requires systemic steroids or immunosuppressive agents

  * NOTE: Exceptions are allowed for:

    * Vitiligo
    * Resolved childhood asthma/atopy
    * Intermittent use of bronchodilators or inhaled steroids
    * Daily steroids at dose of =\< 10mg of prednisone (or equivalent)
    * Local steroid injections
    * Stable hypothyroidism on replacement therapy
    * Stable diabetes mellitus
    * Sjogren's syndrome
* Any prior head or neck chemotherapy, radiotherapy, and/or immunotherapy
* Any of the following prior therapies:

  * Live vaccine \< 30 days prior to registration, including intranasal flu vaccine (e.g. Flu-Mist®) (Note: Seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed). Examples of live vaccines include, but are not limited to, the following: measles, mumps, rubella, varicella/zoster (chicken pox), yellow fever, rabies, Bacillus Calmette-Guerin (BCG), and typhoid vaccine. Seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed; however, intranasal influenza vaccines (eg, FluMist) are live attenuated vaccines and are not allowed
  * Chemotherapy or targeted small molecule therapy \< 21 days prior to registration
  * Investigational therapy or investigational device \< 30 days prior to registration
  * Any prior investigational HPV-specific therapeutic vaccine
* Current or prior use of immunosuppressive medication \< 14 days prior to registration

  * The following are exceptions to this criterion:

    * Intranasal, inhaled, topical steroids, or local steroid injections (e.g., intraarticular injection)
    * Systemic corticosteroids at physiologic doses not to exceed 10 mg/day of prednisone or its equivalent
    * Steroids as premedication for hypersensitivity reactions (e.g., premedication for computed tomography \[CT\] scans)
* Uncontrolled intercurrent illness including, but not limited to:

  * Ongoing or active infection requiring systemic therapy
  * Interstitial lung disease
  * Serious, chronic gastrointestinal conditions associated with diarrhea (e.g., Crohn's disease or others)
  * Known active hepatitis B (i.e., known positive HBV surface antigen (HBsAg) reactive)
  * Known active hepatitis C (i.e., positive for HCV ribonucleic acid \[RNA\] detected by polymerase chain reaction \[PCR\])
  * Known human immunodeficiency virus (HIV) (Note: Patients on stable highly active antiretroviral therapy (HAART) for \>= 6 weeks with CD4 counts \>= 200 cells/mm\^3 undetectable HIV viral load by quantitative PCR and no opportunistic infections Castlemaan's Disease =\< 12 months prior to enrollment are allowed)
  * Known active tuberculosis (TB)
  * Symptomatic congestive heart failure
  * Unstable angina pectoris
  * Unstable cardiac arrhythmia or
  * Psychiatric illness/social situations that would limit compliance with study requirements (e.g., substance abuse)
* History of allogeneic hematopoietic transplant or any solid organ transplant
* Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm
* Other active malignancy \< 2 years prior to registration

  * EXCEPTIONS: Non-melanotic skin cancer (SCC/BCC), micropapillary thyroid cancer, Gleason 6 prostate cancer, carcinoma-in-situ of the breast or cervix
* Any of the following conditions =\< 6 weeks prior to registration:

  * Cerebrovascular accident (CVA)
  * Admission for unstable angina
  * Cardiac angioplasty or stenting or coronary artery bypass graft surgery
  * Untreated pulmonary embolism or untreated deep venous thrombosis (DVT)
  * Arterial thrombosis
* Receipt of immunotherapy/immunomodulatory or immunosuppressive agents (e.g., IFNs, tumor necrosis factor, interleukins, immunoglobulins or other biologic response modifiers \[GM-CSF, GCSF\] =\< 6 weeks prior to registration