Intralesional Influenza Vaccine for the Treatment of Stage I-IV Melanoma

Inclusion Criteria: * Males or females * 18 to 99 years of age * Histologically confirmed cutaneous melanoma by historical pathology report review, clinical Stage I-III (Cohort #1), or Stage IV (Cohort #2) cutaneous melanoma * At least one, biopsy-proven, palpable melanoma tumor deposit suitable for intralesional injection measuring ≥ 1 cm by digital caliper (with digital photography documentation) or ultrasound (with ultrasound image documentation) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0-1 * Absolute neutrophil count (ANC) \>= 1.5 x 10\^3/mm\^3 (drawn at or not more than 30 days prior to the screening visit) * Hemoglobin (Hgb) \>= 8 g/dL (drawn at or not more than 30 days prior to the screening visit) * Platelet count \>= 100 x 10\^3/mm\^3 (drawn at or not more than 30 days prior to the screening visit) * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 2.5 x upper limit of normal (ULN) or =\< 5 x ULN in patients with liver metastases (Cohort 2 only) (drawn at or not more than 30 days prior to the screening visit) * Prothrombin time =\< 1.5 x ULN (drawn at or not more than 30 days prior to the screening visit) * Total bilirubin =\< 1.5 x ULN (unconjugated bilirubin of \< 3 x ULN for patients with known Gilbert syndrome) (drawn at or not more than 30 days prior to the screening visit) * Creatinine clearance of \>= 50 ml/min by Cockcroft-Gault equation (drawn at or not more than 30 days prior to the screening visit) * Women of childbearing potential (WOCBP) must agree to use effective contraceptive methods from screening until at least: * Cohort 1: 14 days after the surgical resection for subjects in Cohort 1 * Cohort 2: * Nivolumab: 5 months after the last dose of either nivolumab or intralesional Flucelvax, whichever is later * Pembrolizumab: 4 months after the last dose of either pembrolizumab or intralesional Flucelvax, whichever is later * Ipilimumab: 3 months after the last dose of either ipilimumab or intralesional Flucelvax, whichever is later * Relatlimab + nivolumab (marketed under the trade name Opdualag): 5 months after the last dose of either Opdualag or intralesional Flucelvax, whichever is later. * Combination ipilimumab with other checkpoint inhibitor: Whichever is later: * 3 months after the last dose of either ipilimumab or intralesional Flucelvax * Above-bulleted recommendation for nivolumab or pembrolizumab * Non-childbearing potential is defined as a woman who meets either of the following criteria: a) postmenopausal state defined as no menses for 12 months without an alternative medical cause, or b) documented hysterectomy, bilateral tubal ligation, or bilateral oophorectomy * Effective contraception methods are defined as one of the following: * True abstinence, defined as refraining from heterosexual intercourse, when this is in line with the preferred and usual lifestyle of the subject * Periodic abstinence (e.g., calendar, ovulation, symptothermal, post-ovulation methods), declaration of abstinence for the duration of a trial, and withdrawal are not acceptable methods of contraception * Condoms and spermicide * Diaphragm and spermicide * Oral or implanted hormonal contraceptive * An intra-uterine device * WOCBP must have a negative pregnancy test (serum or urine) Exclusion Criteria: * Known allergy or intolerance to influenza vaccination * Subjects with condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone/equivalent) or other immunosuppressive medications within 14 days of study drug administration * Active, known or suspected autoimmune disease * Active brain metastasis or leptomeningeal metastasis * Diagnostic biopsy of ocular or mucosal melanoma * Any melanoma therapy within 6 months of enrollment; though prior surgical resection is permitted * Incarcerated patients * Patients known to be HIV positive are eligible if they meet the following criteria within 30 days prior to randomization: stable and adequate CD4 counts (≥ 350 mm\^3), and serum HIV viral load of \< 25,000 IU/ml. Patients may be on or off anti-viral therapy so long as they meet the CD4 count criteria * Pregnant or lactating patients * Patients incapable of independently providing consent