Brief Summary: This phase I trial investigates the effects of influenza vaccine in treating patients with stage I-IV melanoma. While intramuscular administration of influenza vaccine provides immunization against the influenza virus, giving influenza vaccine directly into the tumor (intralesional) may decrease the size of the injected melanoma tumor, or the extent of the melanoma within the body.

Detailed Description: PRIMARY OBJECTIVE: I. To evaluate the safety and tolerability and determine the maximum tolerated dose of intralesional (quadrivalent inactivated influenza vaccine (unadjuvanted influenza vaccine) for patients with a) resectable melanoma as monotherapy, and b) metastatic melanoma, concurrent with standard of care (single- or dual-agent) checkpoint inhibition. SECONDARY OBJECTIVES: I. To evaluate tumor dimensions of injected (Cohorts #1-2) and non-injected lesions (Cohort #2 only), by caliper or ultrasound measurement. (Clinical endpoint) II. To determine time to disease progression (local or distant). (Clinical endpoint) III. To evaluate immunohistochemistry density, cells/mm\^2: CD4, CD8, PD-L1, PD1, CD56, CD20, CD45RO, FOXP3. (Tumor-based endpoint) IV. To evaluate granzyme B H-score. (Tumor-based endpoint) V. To evaluate NanoString Pan Cancer Immune Profiling Panel. (Tumor-based endpoint) VI. To evaluate tumor-infiltrating lymphocytes: not identified, present (non-brisk), present (brisk), cannot be determined. (Tumor-based endpoint) VII. To evaluate degree of tumor regression (percent). (Tumor-based endpoint) VIII. To evaluate changes in micro ribonucleic acid (microRNA) expression. (Tumor-based endpoint) IX. To evaluate of flow cytometry for T-cell subset evaluation and changes in circulating microRNA. (Blood draw endpoint) EXPLORATORY OBJECTIVE: I. To evaluate the evidence of immunologic activation in blood and tissue specimens. OUTLINE: This is dose-escalation study. Patients are assigned to 1 of 2 cohorts. COHORT I: Patients receive quadrivalent inactivated influenza vaccine intramuscularly (IM) on day 0 and intratumorally on days 2 and 14 in the absence of disease progression or unacceptable toxicity. Patients then undergo surgery on day 28. COHORT II: Patients receive quadrivalent inactivated influenza vaccine IM on day 0 and intratumorally on days 2, 14, 28, 42, 56, 70, 84, and 98 in the absence of disease progression or unacceptable toxicity. Patients also receive standard of care ipilimumab, nivolumab, pembrolizumab, or Opdualag. After completion of study treatment, patients are followed up for up to 1 year.