TMV Vaccine Alone or With Toripalimab-tpzi for Recurrent Head and Neck Squamous Cell Carcinoma

Inclusion Criteria: * Age 18 years or older. * Recurrent, histologically proven squamous cell carcinoma of the head and neck (HNSCC), amenable to curative salvage surgery. EBV- and HPV/p16-positive and -negative tumors are allowed. Squamous cell carcinoma of the oral cavity, larynx, hypopharynx, oropharynx, sinonasal carcinoma, and cancer of unknown primary (squamous cell carcinoma only) are allowed. * Oropharyngeal tumors must have EBV and p16 or HPV testing done. * Tumor tissue must be available and banked (below -70°C) at the time of salvage surgery (Phase 1a) or biopsy (Phase 1b), with the tissue collection informed consent signed. * Enough tissue collected at salvage surgery (Phase 1a) or biopsy (Phase 1b) to make at least 3 doses of vaccine: resectable tumor tissue weighing at least 1 gram (approximately 0.5 to 3 cm³) with adequate cellularity (greater than 40% viable tumor cellularity), as assessed by pathology using the corresponding biopsy or surgical specimen. * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. * Phase 1b only: Tumor must be PD-L1 positive (combined positive score \[CPS\] of 1 or higher) by IHC (22C3), performed in the MD Anderson CLIA-certified laboratory. * Phase 1b only: Measurable disease per RECIST 1.1. * Adequate hematologic function: * Absolute neutrophil count ≥ 1,500 cells/µL * Platelets ≥ 100,000/µL * Hemoglobin ≥ 9.0 g/dL (red blood cell transfusion allowed) * Adequate hepatic function: * Total bilirubin ≤ 1.5 x upper limit of normal (ULN) * AST and ALT ≤ 2.5 x ULN * Albumin ≥ 3.0 g/dL * Adequate renal function: * Serum creatinine ≤ 1.5 x ULN * Creatinine clearance ≥ 50 mL/min, calculated using the Cockcroft-Gault formula * INR ≤ 1.5. Anticoagulation is allowed. Patients receiving low molecular weight heparin on a stable therapeutic dose for more than 2 weeks, or with a factor Xa level \< 1.1 U/mL, are allowed. * Willing and able to comply with scheduled visits, treatment plans, laboratory tests, and other study procedures. * Able to understand and willing to sign written informed consent documents. * Females of childbearing potential (FCBP) must agree to use 2 acceptable methods of contraception, 1 from Category A and 1 from Category B, during study treatment and for 3 months after the last dose, or agree to remain abstinent from heterosexual intercourse when this is consistent with their preferred and usual lifestyle. * Category A (highly effective methods): hormonal contraceptives (oral, injectable, transdermal, or implantable); intrauterine device (IUD) or intrauterine hormone-releasing system; vasectomized partner (with confirmed azoospermia) as the sole sexual partner * Category B (barrier methods): male condom; female condom; diaphragm or cervical cap used with spermicide * An FCBP is a sexually mature woman who has not had a hysterectomy or bilateral oophorectomy, or who has not been naturally postmenopausal for at least 24 consecutive months (has had menses during the preceding 24 consecutive months). * Males who are sexually active with an FCBP must agree to use an effective method of contraception (for example, a male condom) during study treatment and for 3 months after the last dose, in addition to their partner's contraception, or agree to remain abstinent from heterosexual intercourse when this is consistent with their preferred and usual lifestyle. * Females of childbearing potential must have a negative serum pregnancy test at study entry. Exclusion Criteria: * Nasopharyngeal carcinoma, salivary gland tumors, or non-squamous cell histology of the head and neck. * Other malignancies within the past 5 years, other than those specified in the inclusion criteria. Patients with prior malignancies, including pelvic cancer, are eligible if disease free for more than 5 years. Patients with prior in situ carcinomas are eligible if there was complete removal. * Not enough tissue collected at surgery (Phase 1a) or biopsy (Phase 1b) for the planned 3 doses (resectable tumor tissue weighing less than 1.0 gram). * Treatment with chronic immunosuppressants (for example, cyclosporine following transplantation). * Prior organ allograft or allogeneic bone marrow transplantation. * Active, known, or suspected autoimmune disease. Patients with vitiligo, psoriasis not requiring systemic treatment, alopecia, resolved childhood asthma/atopy, type I diabetes mellitus, residual hypothyroidism due to an autoimmune condition requiring only hormone replacement, or conditions not expected to recur in the absence of an external trigger are permitted. * A condition requiring systemic treatment with corticosteroids (\> 10 mg daily prednisone equivalent) or other immunosuppressive medications within 14 days of study drug administration. Inhaled or topical steroids and adrenal replacement doses \> 10 mg daily prednisone equivalent are permitted in the absence of active autoimmune disease. * Uncontrolled intercurrent illness including, but not limited to, HIV-positive patients receiving combination antiretroviral therapy, ongoing or active infection, symptomatic congestive heart failure (NYHA Class III or IV), unstable angina pectoris, ventricular arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Other medications, severe acute or chronic medical or psychiatric conditions, or laboratory abnormalities that may increase the risk of study participation or study drug administration, or may interfere with interpretation of study results, and in the judgment of the investigator would make the patient inappropriate for the study. * Clinical evidence of bleeding diathesis or coagulopathy. * Active bacterial or fungal infection requiring systemic treatment within 7 days of treatment. * Use of other investigational drugs within 28 days or at least 5 half-lives (whichever is longer) before study drug administration. * Non-oncology vaccines within 28 days before planned treatment. * Failure to reco