Sonidegib and Pembrolizumab in Treating Patients With Advanced Solid Tumors

Inclusion Criteria: * Age \>= 18 years * Measurable disease by RECIST criteria. * Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 0 or 1. * Hemoglobin \>= 9.0 g/dL (obtained =\< 28 days prior to registration). * Absolute neutrophil count (ANC) \>= 1000/mm\^3 (obtained =\< 28 days prior to registration). * Platelet count \>= 100,000/mm\^3 (obtained =\< 28 days prior to registration). * Total bilirubin =\< 1.5 x upper limit of normal (ULN) OR direct bilirubin =\< ULN (obtained =\< 28 days prior to registration). * Alanine aminotransferase (ALT) and aspartate transaminase (AST) =\< 2.5 x ULN (=\< 5 x ULN for patients with liver involvement) (obtained =\< 28 days prior to registration). * Creatinine phosphokinase (CK) =\< 2.5 x ULN (obtained =\< 28 days prior to registration). * Serum creatinine =\< 1.5 x ULN or calculated creatinine clearance \>= 50 ml/min using the Cockcroft-Gault formula (obtained =\< 28 days prior to registration). * Negative serum pregnancy test done =\< 7 days prior to registration, for persons of childbearing potential only. * Patients of childbearing potential agree to use two forms of medically approved contraception while taking the study drug and for 20 months following the last dose of study drug. Patients with partners of childbearing potential agree to use condoms, even after vasectomy, to avoid potential drug exposure to partner during study drug and for 8 months following the last dose of study drug. * Willing to return to enrolling institution for follow-up (during the active monitoring phase of the study). * Willing to provide blood samples for correlative research purposes. * Must be able to swallow capsules and have no significant impairment in gastrointestinal absorption. * Willing and able to provide informed consent. * PART A (DOSE ESCALATION): Patient must satisfy all subsets in one of the following: * Patients with NSCLC. * Pathologically confirmed metastatic non-small cell lung cancer (NSCLC). * Patients with EGFR, ALK, or BRAF genomic abnormalities must have also received and progressed on prior Food and Drug Administration (FDA)-approved targeted therapies * Melanoma. * Unresectable or metastatic melanoma. * NOTE: Previous treatment using PD-1/PD-L1 checkpoint inhibitors is allowed. * Head and neck squamous cell cancer (HNSCC): * Recurrent or metastatic HNSCC with disease progression on or after prior platinum-containing chemotherapy. * NOTE: Previous treatment using PD-1/PD-L1 checkpoint inhibitors is allowed. * Urothelial carcinoma (locally advanced or metastatic). * Newly diagnosed cisplatin ineligible patients. OR * Progression during or within 12 months of treatment with platinum-containing agent. * Microsatellite instability-high (MSI-H) cancer. * Unresectable or metastatic solid tumors that progressed on prior treatment and are MSI-H or mismatch repair deficient. * No satisfactory alternative treatment options available. For colorectal cancer, must have progressed following treatment with fluoropyramidine, oxaliplatin, and irinotecan. * Gastric or gastroesophageal junction adenocarcinoma * Locally advanced or metastatic tumors that express PD-L1 as evidenced by a combined positive score (\>= 1) using the PD-L1 immunohistochemistry (IHC) 223C pharmDx test (Dako). * Disease progression on 2 or more prior systemic therapies. * PART B (DOSE EXPANSION) COHORT A: Recurrent or metastatic HNSCC * Pathologically confirmed recurrent or metastatic HNSCC * Disease progression on or after platinum-containing chemotherapy * NOTE: Previous treatment using PD-1/PD-L1 checkpoint inhibitors is allowed. * PART B ( DOSE EXPANSION) COHORT B: Refractory NSCLC. * Pathologically confirmed metastatic non-small cell lung cancer (NSCLC). * Disease progression on \>= 1 prior line of systemic therapy. * NOTE: Previous treatment using PD-1/PD-L1 checkpoint inhibitors is allowed. * Patients with EGFR, ALK, or BRAF genomic abnormalities must have also received and progressed on prior FDA-approved targeted therapies. Exclusion Criteria: * Any of the following because this study involves an agent that has known genotoxic, mutagenic and teratogenic effects: * Pregnant persons. * Nursing persons. * Persons of childbearing potential and with partners of childbearing potential who are unwilling to employ adequate contraception. * CTCAE \>= grade 3 treatment-emergent adverse event (TEAE) to prior checkpoint inhibitor, TEAE requiring systemic corticosteroids, or permanent treatment discontinuation due to toxicity. * Neuromuscular disorders (e.g. inflammatory myopathies, muscular dystrophy, amyotrophic lateral sclerosis and spinal muscular atrophy), or a history of rhabdomyolysis. * Concomitant treatment with drugs that are recognized to cause rhabdomyolysis, including statins. * NOTE: Patients taking such medications need to be discontinued at least 2 weeks prior to starting sonidegib treatment. If an agent to control lipids is required, pravastatin may be given with caution. * Receiving strong inhibitors or inducers of CYP3A4/5, moderate inducers of CYP3A4, and/or grapefruit/grapefruit juice or starfruit products that cannot be discontinued before starting treatment with sonidegib. NOTE: Medications that are strong CYP3A4/5 inhibitors or inducers, moderate inducers of CYP3A4, and grapefruit/grapefruit juice/starfruit products should be discontinued at least 4 weeks prior to starting treatment with sonidegib. * Active autoimmune diseases that have required systemic treatment modifications within the past 3 months or that require chronic systemic steroids or immunosuppressive agents. * Requirement for systemic corticosteroids (\> 10 mg daily prednisone equivalent) or other immunosuppressive medications =\< 14 days prior to registration. NOTE: Inhaled or topical steroids, and adrenal replacement steroid doses \>10 mg daily prednisone equivalent, are permitted in the absence of active autoimmune disease. * Life expectancy \< 3 months. * Central nervous system metastases that are untreated, symptomatic, or require steroids. NOTE: Patients with history of stable treated brain metastases are eligible. Stable treated metastases are defined as follows: * No evidence of progression for \>= 8 weeks on brain imaging (either magnetic resonance imaging \[MRI\] or computed tomography \[CT\] scan). * No corticosteroid use for brain metastases for \>= 2 weeks before randomization. * \>= 8 weeks from completion of definitive treatment for brain metastases. * Any of the following prior therapies: * Major surgery =\< 4 weeks prior to registration. * Received any experimental drugs or anti-neoplastic therapy =\< 4 weeks prior to receiving the first dose of study treatment * Received a live vaccine =\< 30 days prior to registration * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens. * Ongoing AE due to prior treatment not recovered to =\< grade 1 per CTCAE or baseline unless clinically nonsignificant and/or stable with supportive therapy