A Vaccine (CDX-1401) With or Without a Biologic Drug (CDX-301) for the Treatment of Patients With Stage IIB-IV Melanoma

Inclusion Criteria: * Patients with fully resected stage IIb through IV melanoma, with melanoma validated by histology or cytology, who have NOT received prior therapy. * Patients may have had primary cutaneous, mucosal, or ocular melanoma or metastasis from an unknown primary site. * Tissue should be submitted for evaluation of NY-ESO-1 expression and T-cell infiltrates. However, availability of tissue and/or positivity for NY-ESO-1 is not mandatory. * Prior radiation, chemotherapy or biologics NOT allowed * Not currently receiving any anticancer therapy * Age \>= 18 years * Because no dosing or adverse event (AE) data are currently available on the use of CDX-1401 or CDX-301 in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials. * Eastern Cooperative Oncology Group (ECOG) performance score of 0-1 * Life expectancy of at least 6 months * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,000/mcL * Platelets \>= 75,000/mcL * Hemoglobin \> 9 g/dL * Total bilirubin \< 1.5 x institutional upper limit of normal (bilirubin \< 3 x institutional upper limit of normal for Gilbert's syndrome) * Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT)(serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Creatinine \< 1.5 x institutional upper limit of normal OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * The first six patients enrolled in the Flt3L arm of the study cannot be human immunodeficiency virus (HIV)-positive. After the evaluation of safety in the first 6 patients, HIV-positive patients with adequate immune function as evidenced by stable CD4 counts \>= 350/mm\^3 are allowed to participate if the following criteria are met: * maintained on stable antiretroviral therapy with no significant drug interactions, and * no recent history of acquired immunodeficiency syndrome (AIDS) indicator conditions (\> 2 years from enrolling in trial), and * physician providing patient's care for HIV must also approve of patient entering the study * Females of childbearing potential must have a negative pregnancy test within 7 days before the initiation of protocol therapy. * The effects of CDX-1401 or CDX-301 on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control or abstinence) before study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception before the study, for the duration of study participation, and 4 months after completion of CDX-1401 or CDX-301 administration. * NOTE: Subjects are considered not of child-bearing potential if they are surgically sterile, they have undergone a hysterectomy, bilateral tubal ligation, or bilateral oophorectomy, or they are postmenopausal. Menopause is the age associated with complete cessation of menstrual cycles, menses, and implies the loss of reproductive potential. By a practical definition, it assumes menopause after 1 year without menses with an appropriate clinical profile at the appropriate age. * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had cytotoxic chemotherapy, radiotherapy, interferon (IFN), or ipilimumab before entering the study * Immunosuppressive therapy within 30 days prior to initiation of protocol therapy * Steroid therapy, or steroid therapy with more than 7 consecutive days of steroids within the prior 4 weeks * The use of prednisone or equivalent \< 0.125 mg/kg/day (absolute maximum of 10 mg/day) as replacement therapy is permitted * Inhaled or topical corticosteroids are permitted * Patients who are receiving any other investigational agents * Current or history of systemic autoimmune disease requiring systemic therapy. * NOTE: The following will not be exclusionary: * The presence of laboratory evidence of autoimmune disease (e.g., positive antinuclear antibody \[ANA\] titer) without associated symptoms * Clinical evidence of vitiligo * Other forms of depigmenting illness * Cardiovascular disease that meets one of the following: congestive heart failure (New York Heart Association Class III or IV), active angina pectoris, or recent myocardial infarction (within the last 6 months) * Cirrhosis or chronic hepatitis C virus positivity or chronic hepatitis B infection * NOTE: A positive hepatitis B serology indicative of previous immunization (i.e., hepatitis B virus surface antibody \[HBsAb\]-positive and hepatitis B virus core antibody \[HBcAb\]-negative), or a fully resolved acute hepatitis B virus infection is not an exclusion criterion * Known history of immunodeficiency disorder other than HIV-positive status * Extensive active brain disease including symptomatic brain metastases or presence of leptomeningeal disease * NOTE: Patients with brain metastasis, after definitive therapy with surgery or stereotactic radiation and stable off steroids for \>= 4 weeks, are eligible * Other invasive cancers that are clinically active * Pregnancy or nursing or unwilling to take adequate birth control during therapy * NOTE: Pregnant women are excluded from this study because CDX-1401 or CDX-301 and poly-ICLC have an unknown potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with CDX-1401 or CDX-301, breastfeeding should be discontinued if the mother is treated with CDX-1401 or CDX-301 and poly-ICLC * History of allergic reactions attributed to compounds of similar chemical or biologic composition to CDX-1401 or CDX-301 or poly-ICLC * Prior organ allograft or allogeneic transplantation, if the transplanted tissue is still in place * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Medical or psychiatric illness that would, in the opinion of the investigator, preclude participation in the study or the ability of patients to provide informed consent for themselves * History of pulmonary disease such as emphysema or chronic obstructive pulmonary disease (COPD) (forced expiratory volume in 1 second \[FEV1\] \< 60% of predicted for height and age). Pulmonary function tests (PFTs) are required in patients with prolonged smoking history or symptoms of respiratory dysfunction * Vaccinations other than those given as part of this research study (with the exception of influenza vaccine) are prohibited throughout the duration of study participation. * NOTE: Influenza vaccination (inactivated) is permitted during the flu season. The preferred time is 7 to 14 days after CDX-1401 administration