Pembrolizumab and Vorinostat in Treating Patients With Recurrent Squamous Cell Head and Neck Cancer or Salivary Gland Cancer That Is Metastatic and/or Cannot Be Removed by Surgery

Inclusion Criteria: * Phase I run in: biopsy proven RMHNSCC with the following primary sites: nasopharynx, paranasal sinus, nasal cavity, skin/cutaneous sites; patients with unknown head and neck primary sites will be enrolled; patients with recurrent or metastatic squamous cell carcinomas of the head and neck (regardless of primary site) who are either unwilling to receive or have contraindications (deemed by treating physician) to standard systemic chemotherapy will also be eligible; patients with biopsy proven RMSGC be eligible as well * Phase II expansion: biopsy proven RMHNSCC, of any primary site (including unknown primary) and RMSGC will be eligible * Have evidence of disease progression by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria within 3 months prior to study enrollment; if the patient was receiving a prior line of systemic therapy, he/she should have evidence of disease progression on that line of treatment prior to enrollment * Have received any number lines of prior systemic therapy (including systemic therapy in the curative intent setting) * Be willing and able to provide written informed consent for the trial and comply with the study visit requirements * Have measurable disease based on RECIST 1.1 * Have provided tissue from an archival tissue sample or newly obtained core or excisional biopsy of a tumor lesion * Have a performance status of 0 or 1 on the Eastern Cooperative Oncology Group (ECOG) performance scale * Absolute neutroph…