Testing the Addition of an Individualized Vaccine to Durvalumab and Tremelimumab and Chemotherapy in Patients With Metastatic Triple Negative Breast Cancer

Inclusion Criteria: * Patients must have a histologically confirmed diagnosis of metastatic invasive triple negative breast cancer. Patients with clinical and/or radiologic suspicion of metastatic TNBC can be consented prior to this confirmation. * Estrogen receptor (ER) and progesterone receptor (PR) less than Allred score of 3 OR less than 1% positive staining cells in the invasive component of the tumor. * HER2 negative by fluorescence in situ hybridization (FISH) or immunohistochemistry (IHC) staining 0 or 1+. * PD-L1 negative by a Clinical Laboratory Improvement Act (CLIA) approved laboratory using compatible assays appropriate for treatment decisions. * Patients may have measurable or evaluable disease. * Patients must be willing to undergo biopsy and have accessible lesions for a new biopsy, or they must have sufficient tissue available from a biopsy performed for standard of care (specifications below). If patient does not have enough archived tissue available, a new biopsy is required. A tumor specimen obtained from relapsed primary, metastatic, or locally advanced sites of disease (if applicable) must be submitted. Acceptable samples include core needle biopsies for deep tumor tissue (minimum 4 cores) or excisional, incisional, punch, or forceps biopsies for cutaneous, subcutaneous, or mucosal lesions. Formalin-fixed, paraffin-embedded (FFPE) tumor specimens in paraffin blocks are preferred; FFPE tumor tissue sections on slides may be provided if sufficient material (15 x 10μ, unstained) is available. Fine-needle aspiration, brushing, cell pellet from pleural effusion, bone metastases, and lavage samples are not acceptable. * No prior therapy for metastatic TNBC. Patients who have received taxane-based adjuvant therapy are required to have a disease-free interval of at least 12 months after completion of taxane therapy. * Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of durvalumab (MEDI4736) and tremelimumab in combination with neoantigen vaccine 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 status =\< 1 (Karnofsky \>= 60%). * Body weight \> 30 kg. * Must have a life expectancy of at least 12 weeks. * Absolute neutrophil count \>= 1,500/mcL. * Platelets \>= 100,000/mcL. * Hemoglobin \>= 9.0 g/dL. * Serum bilirubin =\< 1.5 x institutional upper limit of normal. * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal unless liver metastases are present, in which case it must be =\< 5 x institutional upper limit of normal. * Calculated creatinine clearance \> 40 mL/min by the Cockcroft-Gault formula or by 24-hour urine collection for determination of creatinine clearance. * Evidence of post-menopausal status or negative urinary or serum pregnancy test for female pre-menopausal patients. Women will be considered post-menopausal if they have been amenorrheic for 12 months without an alternative medical cause. The following age-specific requirements apply: * Women \< 50 years of age would be considered post-menopausal if they have been amenorrheic for 12 months or more following cessation of exogenous hormonal treatments and if they have luteinizing hormone and follicle-stimulating hormone levels in the post-menopausal range for the institution or underwent surgical sterilization (bilateral oophorectomy or hysterectomy). * Women \>= 50 years of age would be considered post-menopausal if they have been amenorrheic for 12 months or more following cessation of all exogenous hormonal treatments, had radiation-induced menopause with last menses \> 1 year ago, had chemotherapy-induced menopause with last menses \> 1 year ago, or underwent surgical sterilization (bilateral oophorectomy, bilateral salpingectomy or hysterectomy). * The effects of durvalumab (MEDI4736) and tremelimumab and neoantigen vaccine on the developing human fetus are unknown. For this reason and because these agents may be teratogenic, women of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and through 180 days after completion of durvalumab (MEDI4736) and tremelimumab. 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. * Patient is willing and able to comply with the protocol for the duration of the study including undergoing treatment and scheduled visits and examinations including follow up. * Human immunodeficiency virus (HIV)-positive patients are eligible provided they have a negative viral load, CD4 count \> 250, and are on a stable antiretroviral regimen. * Ability to understand and the willingness to sign a written informed consent document. Patients with impaired decision-making capacity who have a close caregiver or legal guardian are also eligible with the consent of the caregiver/guardian. Exclusion Criteria: * Patients who are not considered to be candidates for carboplatin + gemcitabine for first line therapy of their metastatic triple negative breast cancer are not eligible. * Patients who have had chemotherapy, radiotherapy (to more than 30% of the bone marrow), or biologic therapy within 30 days (42 days for nitrosoureas or mitomycin C) prior to entering the study. * Patients who have received prior immunotherapy for metastatic disease. * Patients who have not recovered from grade \>= 2 adverse events due to prior anti-cancer therapy with the exception of alopecia, vitiligo, and the laboratory values defined in the inclusion criteria. * Patients with grade \>= 2 neuropathy will be evaluated on a case-by-case basis after consultation with the study physician. * Patients with grade \>= 2 endocrinological adverse events (AEs), (e.g., hypothyroidism, adrenal insufficiency, hypopituitarism, or diabetes mellitus), must have been on a stable dose of supplemental therapy for at least 2 weeks before screening to be eligible for this study, and the endocrinological AE must be stable in the opinion of the treating physician. * Patients who are receiving any other investigational agents or who have received an investigational agent within the last 30 days. * Receipt of live attenuated vaccination within 6 months prior to study entry or within 30 days of receiving durvalumab (MEDI4736) and tremelimumab. * Note: Patients, if enrolled, should not receive live vaccine whilst receiving study treatment and up to 30 days after the last dose of study treatment. * Major surgical procedure within 28 days prior to the first dose of durvalumab (MEDI4736) and tremelimumab. Local surgery of isolated lesions for palliative intent is acceptable. * Current or prior use of immunosuppressive medication within 14 days before the first dose of durvalumab (MEDI4736) or tremelimumab. The following are exceptions to this criterion: * Intranasal, inhaled, topical steroids or local steroid injections (e.g. intra-articular injection) * Systemic corticosteroids at physiological doses which are not to exceed 10 mg/day of prednisone or an equivalent corticosteroid * Steroids as premedication for hypersensitivity reactions (e.g. CT scan premedication) * Steroids for symptoms from brain metastases as defined. * Spinal cord compression or active brain metastases and/or carcinomatous meningitis. Subjects who have had brain metastases resected or have received radiation therapy ending at least 4 weeks prior to registration are eligible if they meet all of the following criteria: * Residual neurological symptoms have resolved to grade =\< 2 * On stable doses of dexamethasone, if applicable and if acceptab