This phase II trial studies how well cisplatin, docetaxel, and pembrolizumab work in treating patients with stage II-III laryngeal cancer. Drugs used in chemotherapy, such as cisplatin and docetaxel, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Immunotherapy with monoclonal antibodies, such as pembrolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Giving cisplatin, docetaxel, and pembrolizumab may help to control the disease.

PRIMARY OBJECTIVES: I. To determine the clinical benefit rate (CBR) of patients with stage II or III larynx squamous cell carcinoma (SCC) after 2 cycles of pembrolizumab, cisplatin and docetaxel (PCD), and the pathologic complete response (pCR) rate after 4 cycles of PCD. SECONDARY OBJECTIVES: I. To determine safety and tolerability of PCD in patients with larynx SCC. II. To determine the laryngeal preservation rate (LPR) at 2 years in the overall population and in the subgroup who achieves a pCR. III. To determine the 2 year relapse-free survival (RFS) and overall survival (OS) in the overall population and in the subgroup who achieves a pCR. IV. To determine patient-reported outcomes (PROs) using M. D. Anderson Symptom Inventory-Head and Neck (MDASI-HN) and swallow function using Dynamic Imaging Grade of Swallowing Toxicity (DIGEST). EXPLORATORY OBJECTIVES: I. To assess predictive tissue and blood-based biomarkers of benefit from PCD in larynx SCC. OUTLINE: Patients receive cisplatin intravenously (IV) over 1 hour, docetaxel IV over 1 hour (patients who develop significant adverse events to cisplatin treatment may receive carboplatin IV over 1 hour instead), and pembrolizumab IV over 30 minutes on day 1. Treatment repeats every 21 days for up to 4 cycles in the absence of disease progression or unacceptable toxicity. Patients who completely respond to the study drugs (the disease appears to go away) then receive pembrolizumab IV over 30 minutes on day 1 for 4 additional cycles in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up at 3 weeks, then every 6-12 weeks for up to 2 years.