DNA Plasmid-encoding Interleukin-12/HPV DNA Plasmids Therapeutic Vaccine INO-3112 and Durvalumab in Treating Patients With Recurrent or Metastatic Human Papillomavirus Associated Cancers

Inclusion Criteria: * Written informed consent in accordance with institutional guidelines. * Are able and willing to comply with all study procedures. * For patients who are not human immunodeficiency virus (HIV) positive, cervical, anal, penile, vulvar, or vaginal cancer positive for HPV-16 and/or HPV-18 by the institutionally approved assay. For patients who are HIV positive, histologically or cytologically confirmed diagnosis of cancer at any site that is positive for HPV-16 and/or HPV-18 by the institutionally approved assay. Tumors may be positive for more than 1 HPV subtype as long as HPV-16 and/or HPV-18 is present. Note: For the first 3 patients, only cervical, vulvar, or vaginal cancers will be enrolled. * Patients with cancer that is refractory to standard therapy, that have either relapsed after standard therapy or has no standard therapy that increases survival by at least three months, and/or that are not curable by salvage approaches including resection and/or re-irradiation * Has measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) with a minimum size of 10 mm by computed tomography (CT) scan, except lymph nodes which must have minimum short axis size of 15 mm (CT scan slice thickness no greater than 5 mm in both cases). Indicator lesions must not have been previously treated with surgery, radiation therapy, or radiofrequency ablation unless there is documented Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1 progression in the lesion after such therapy. * All patients must consent to pre-treatment biopsy of the tumor if it can be done safely (as judged by the investigator) during screening. Week 10 on-treatment biopsies will be required for a minimum 10 patients. After 10 paired biopsies have been obtained then week 10 on-treatment biopsy will be made optional. * World Health Organization (WHO)/Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Hemoglobin \>= 9 g/dL (Note: Note: No Transfusion within 7 days of beginning study treatment. Ongoing growth factor support is acceptable if on a stable dose for the past 56 days), within 28 days of day 0. * Cannot be met with recent blood transfusions or require ongoing growth factor support * Absolute neutrophil count \>= 1,000/mm\^3, within 28 days of day 0. * Cannot be met with recent blood transfusions or require ongoing growth factor support * Platelet count \>= 100,000/mm\^3 and no transfusion in prior 4 weeks, within 28 days of day 0. * Cannot be met with recent blood transfusions or require ongoing growth factor support * Total bilirubin (TBL) =\< 1.5 x upper limit of normal (ULN) except patients with documented Gilbert's syndrome (\> 3 x ULN), within 28 days of day 0. * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 x ULN, within 28 days of day 0. * Serum creatinine =\< 2.0 mg/dL or creatinine clearance \>= 40 mL/min (measured or calculated according to the method of Cockcroft and Gault), within 28 days of day 0. * For human immunodeficiency virus (HIV)+ patients: documented HIV-1 infection with CD4 count \> 200 cells/mm\^3 and viral load \< 75 copies/mL, within 28 days of day 0. Exclusion Criteria: * Any concurrent chemotherapy, investigational product (IP), biologic, or hormonal therapy for cancer treatment; receipt of any investigational or approved anticancer therapy (chemotherapy, targeted therapy, biologic therapy, monoclonal antibodies, etc.) within 21 days or 5 half lives, whichever is shorter, prior to the first dose of MEDI0457; concurrent enrollment in another clinical study, unless it is an observational (non-interventional) clinical study or during the follow-up period of an interventional study. * Major surgical procedure or significant traumatic injury within 28 days before the first dose of study drug or anticipation of the need for major surgery during the course of study treatment. * Any unresolved toxicity (National Cancer Institute Common Terminology Criteria for Adverse Event \[CTCAE\] version 4.03 \[v4.03\]) grade 2 or greater from previous anticancer therapy with the exception of alopecia, and the laboratory values defined in the inclusion criterion 8. Hearing loss of grade 3 or lower and peripheral neuropathy of grade 2 or lower is allowed. Subjects with grade \>= 2 neuropathy will be evaluated on a case-by-case basis after consultation with the study physician. Subjects with irreversible toxicity not reasonably expected to be exacerbated by treatment with durvalumab may be included only after consultation with the study physician. * Current or prior use of immunosuppressive medication within 14 days prior to first study dose, with the exception of intranasal and inhaled corticosteroids or systemic corticosteroids at physiologic doses not to exceed 10 mg/day of prednisone or equivalent. Steroids as premedication for hypersensitivity reactions due to radiographic contrast agents are allowed. * Patients requiring therapeutic anticoagulation and irreversible platelet inhibitors (e.g. clopidogrel, prasugrel, or ticagrelor). Low dose aspirin for cardiac prophylaxis is allowed. * History of primary immunodeficiency. * Patients who have had prior exposure to immune-mediated therapy, including but not limited to prior exposure to T-cell and natural killer cell directed therapy, anti-PD-1, anti-PD-L1, anti-CD137, and anti-CTLA4. * History of allogeneic organ transplantation. * Active or prior documented autoimmune or inflammatory disorders (including inflammatory bowel disease \[e.g. colitis, ulcerative colitis or Crohn's disease\], diverticulitis \[with the exception of diverticulosis\], systemic lupus erythematosus, sarcoidosis syndrome, or Wegener syndrome \[granulomatosis with polyangiitis, Graves' disease, rheumatoid arthritis, hypophysitis, uveitis, etc.\]). The following are exceptions to this criterion: patients with vitiligo or alopecia, patients with hypothyroidism (e.g. following Hashimoto syndrome) stable on hormone replacement, any chronic skin condition that does not require systemic therapy, patients without active disease in the last 5 years may be included but only after consultation with the study physician, and patients with celiac disease controlled by diet alone. * Uncontrolled intercurrent illness, including, but not limited to, ongoing or active infection, uncontrolled hypertension, interstitial lung disease, serious chronic gastrointestinal conditions associated with diarrhea, or psychiatric illness/social situations that would limit compliance with study requirement, substantially increase risk of incurring adverse events, or compromise the ability of the patient to give written informed consent. * Patients with spinal cord compression or a history of leptomeningeal carcinomatosis. At the time of day 1 of the study, patients with central nervous system metastases must have been treated and must be asymptomatic and meet the following criteria. 1. No concurrent treatment, inclusive of, but not limited to, surgery, radiation, and/or corticosteroids. (Note: patients are allowed on systemic steroids unless these are being administered to manage central nervous system metastases); 2. Neurologic stability (lack of signs or symptoms greater than baseline prior to radiotherapy) until the time of dosing of MEDI0457; 3. For radiation treatment, patients must be: at least 14 days between last day of stereotactic radiosurgery or gamma-knife treatment and day 1 of protocol treatment, at least 28 days between last day of whole brain radiation therapy and day 1 of protocol treatment, and/or at least 14 days since last dose of corticosteroids and day 1 of protocol treatment. * Patients with cardiovascular (CV) disease conditions including New York Heart Association class 3 or 4 congestive heart failure, unstable angina pectoris, or clinically important cardiac arrhythmias OR a recent (\< 3 months) CV event, including myocardial infarction, unstable angina pectoris, or stroke. * Mean QT interval corrected for heart rate (QTc) \>= 470 ms calculated from electrocardiogram (ECG) using Fridericia's correction by manual read. * Active tuberculosis (clinical evaluation that includes clinical history, physical examination and radiographic findings, and tuberculosis testing in line with local practice) infection. * Presence of acute or chronic hepatitis B (hepatitis B virus \[HBV\]) or active hepatitis C (hepatitis C virus \[HCV\]). Patients with a past or resolved HBV infection (defined as the presence of hepatitis B core antibody \[anti-HBc\] and absence of HBV surface antigen \[HBsAg\]) are eligible. Patients positive for HCV antibody are eligible only if polymerase chain reaction is negative for HCV ribonucleic acid (RNA). * Receipt of live, attenuated vaccine within 30 days prior to study entry or the first dose of MEDI0457. * Note: patients, if enrolled, should not receive live vaccine during the study and up to 30 days after the last dose of IP. * Other untreated coexisting HIV related malignancies. * History of another primary malignancy except for: malignancy treated with curative intent and with no known active disease \>= 2 years before the first dose of IP and of low potential risk for recurrence, adequately treated non-melanoma skin cancer or lentigo maligna without evidence of disease, or adequately treated carcinoma in situ without evidence of disease. * Pregnant or breastfeeding female patients. * Known allergy or hypersensitivity to study treatment or any of the study drugs excipients. * Any medical condition that, in the opinion of the investigator, would interfere with evaluation of the study treatment or interpretation of patient safety or study results. * Patients with active or prior digestive tract bleeding. * Patients with uncontrolled seizures. * Fewer than two acceptable sites exist for intramuscular (IM) injection and electroporation (EP) between the deltoid and lateral quadriceps muscles. Note: a site for injection/EP is not acceptable if there are tattoos or scars within 2 cm of the proposed injection/EP site or if there is implanted metal within the same limb. Any device implanted in the chest (e.g. cardiac pacemaker or defibrillator) excludes the use of the deltoid muscle on the same side of the body. * Patients who are unable to provide informed consent, are incarcerated, or are unable to follow protocol requirements.