HPV Vaccine, Imiquimod, and Metformin Combination Trial

Inclusion Criteria: * Participants must have histologically confirmed locally advanced or metastatic cervical carcinoma (Stage IB2-IVB), vaginal, or vulvar carcinoma (Stage II-IVB), AND not be considered a primary surgical candidate. Patients offered neoadjuvant therapy may be enrolled if they respond and receive chemoradiation. * Participants must have measurable disease, per Recist criteria. See Section 12 (Measurement of Effect) for the evaluation of measurable disease. Radiological evaluation shall occur within approximately 30 days prior to enrollment initiation and start of radiation. * Participants must have an Eastern Cooperative Oncology Group (ECOG) performance status of ≤ 2 * Participants must be ≥ 18 years of age * Participants must have adequate organ function within 28 days of registration, defined as follows: - Absolute neutrophil count ≥ 1,500/µL - Platelets ≥ 100,000/µL - Hemoglobin ≥ 9 g/dL - Serum creatinine ≤ 1.5 x upper limit of normal (ULN) - Total bilirubin ≤ 1.5 x ULN (≤2.0 in patients with known Gilberts syndrome) OR direct bilirubin ≤ 1 x ULN - Aspartate aminotransferase and alanine aminotransferase ≤ 2.5 x ULN unless liver metastases are present, in which case they must be ≤ 5 x ULN * Participants receiving corticosteroids may continue as long as their dose is stable for at least 4 weeks prior to initiating protocol therapy. * Participant must agree to not donate blood during the study or for 90 days after the last dose of study treatment. * Female participants of childbearing potential must have a negative serum pregnancy test within 14 days prior to registration. Females of non-childbearing potential is defined as follows (by other than medical reasons): - ≥45 years of age and has not had menses for \>1 year, post-hysterectomy, post-bilateral oophorectomy, post external beam radiation of 6 Gy to the pelvis, or post-tubal ligation. * Participants must agree to not breastfeed during the study. * Participants must be able to understand the study procedures and agree to participate in the study by providing written informed consent * Participants must be eligible for chemoradiation treatment in the opinion of the treating investigator. * Participants who are HIV+ must have CD4 counts \>200/dL and demonstrate documented Highly active antiretroviral therapy (HAART) compliance m. Participant must have CT (chest/abdomen/pelvis) or PET-CT, within 56 days of registration. * Participants must be newly diagnosed. * Standard chemoradiation using external beam radiation therapy (EBRT) and brachytherapy is permitted for cervical or vaginal carcinoma, and chemoradiation with EBRT for vulvar carcinoma. A lesion must be readily accessible for intratumoral tumor injection. * ECOG performance status ≤2 (Karnofsky ≥60%, see Appendix A). * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. * Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression. * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. Exclusion Criteria: * Patients who are receiving any other investigational agents. * Patients who have untreated, new or progressive brain metastases or leptomeningeal disease. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to agents used in study. * Patients with uncontrolled intercurrent illness. * Patients with psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant women are excluded from this study because cervical carcinoma or vulva carcinoma patients have undergone treatment rendering the patient infertile. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with cervical carcinoma or vulva carcinoma, breastfeeding should be discontinued.