This phase I trial studies the side effects and best dose of TLR8 Agonist VTX-2337 when given together with cetuximab in treating patients with locally advanced, recurrent, or metastatic squamous cell cancer of the head and neck (SCCHN). Biological therapies, such as TLR8 Agonist VTX-2337 may stimulate the immune system in different ways and stop tumor cells from growing. Monoclonal antibodies, such as cetuximab, can block tumor growth in different ways. Some block the ability of tumor cells to grow and spread. Others find tumor cells and help kill them or carry tumor-killing substances to them. Giving TLR8 Agonist VTX-2337 together with cetuximab may kill more tumor cells.