A Vaccine (MV-s-NAP) for the Treatment of Patients With Invasive Metastatic Breast Cancer

Inclusion Criteria: * Age \>= 18 years * COHORT 1 ONLY: Pathologically confirmed invasive breast adenocarcinoma with documented estrogen receptor (ER)/progesterone receptor (PR) /HER2 status and radiographic evidence of distant metastatic disease * COHORTS 2 \& 3 ONLY: Pathologically confirmed invasive breast adenocarcinoma with documented ER/PR/HER2 status and radiographic evidence of distant metastatic or recurrent disease * COHORT 1 ONLY: Radiographic evidence of distant metastatic disease (using 7th edition American Joint Committee on Cancer \[AJCC\] criteria) with two discrete sites of measurable disease * COHORTS 2 \& 3 ONLY: Radiographic evidence of distant metastatic or recurrent disease (using 8th edition AJCC criteria) with at least one site of measurable disease * Prior therapies: * Patients with ER/PR positive, HER2 negative breast cancer must have progressed through at least one prior cytotoxic regimen for advanced disease and no longer be candidates for standard endocrine therapy or combination of endocrine therapy with other agents such as CDK4/6 inhibitors * Patients with HER2 positive breast cancer irrespective of ER/PR status must have received or no longer be candidates for HER2 directed therapy with trastuzumab or pertuzumab * Patients with ER/PR/HER2 negative breast cancer must have progressed through at least one prior cytotoxic regimen for advanced disease * COHORT 1: At least one site of recurrent/metastatic disease that measures \> 1 cm in greatest dimension (\> 2 cm for lung lesions) and is amenable to safe percutaneous intratumoral administration of MV-s-NAP as determined by an interventional radiologist * COHORTS 2 \& 3 ONLY: At least 1 site of recurrent/metastatic disease measuring \> 1 cm in greatest dimension \[\> 2 cm for lung lesions\] (Note that if the lesion injected in cycle 1 is not amenable to re-injection, another lesion could be selected for injection * Absolute neutrophil count (ANC) \>= 1500/uL (=\< 7 days prior to registration) * Platelets (PLT \>= 100,000/uL) (=\< 7 days prior to registration) * Total bilirubin =\< institutional upper limit of normal (=\< 7 days prior to registration) * Aspartate aminotransferase (AST) =\< 2 x upper limit of normal (ULN) (=\< 7 days prior to registration) * Creatinine =\< 1.5 x ULN (=\< 7 days prior to registration) * Hemoglobin \>= 9.0 g/dL (=\< 7 days prior to registration) * Negative pregnancy test done =\< 7 days prior to registration (for women of childbearing potential only) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * Ability to provide informed written consent * Willingness to return to the Mayo Clinic enrolling institution for follow-up * Willingness to provide biologic samples for correlative research purposes * Life expectancy \>= 12 weeks * Concomitant administration of a bone modifying agent (e.g., zoledronic acid or denosumab) is permitted for the prevention or management of skeletal related events in patients with bone metastases and documentation of tolerability with prior exposures Exclusion Criteria: * Known standard therapy for the patient's disease that is potentially curative or definitely capable of extending life expectancy * Clinical or radiographic suspicion of impending visceral crisis due to invasion or compression by tumor * Active infection =\< 5 days prior to registration * History of other malignancy =\< 5 years except for non-melanoma skin cancer or carcinoma in situ of the cervix * Any of the following prior therapies: * Chemotherapy =\< 3 weeks prior to registration * Immunotherapy =\< 4 weeks prior to registration * HER2 directed therapy =\< 3 weeks prior to registration * Targeted therapy =\< 2 weeks prior to registration (e.g., CDK4/6 inhibitors, everolimus) * Investigational agent =\< 4 weeks prior to registration * Any viral or gene therapy prior to registration * Failure to fully recover from acute, reversible effects of prior systemic therapy regardless of interval since last treatment * New York Heart Association classification III or IV, known symptomatic coronary artery disease, or symptoms of coronary artery disease on systems review, or known cardiac arrhythmias (atrial fibrillation or supraventricular tachycardia \[SVT\]) * Untreated or progressive central nervous system (CNS) metastases * NOTE: Patients with a history of treated brain metastases (surgical resection, whole brain radiation, and/or stereotactic radiosurgery) are eligible only if they are asymptomatic and have stable MRI scans for 3 consecutive months, including \< 28 days of study entry * Standing requirement for blood product support * Human immunodeficiency virus (HIV) positive test result or history of other immunodeficiency * History of organ transplantation * History of chronic hepatitis B or C * Other concurrent chemotherapy, immunotherapy, radiotherapy, or any ancillary therapy considered investigational (utilized for a non-Food and Drug Administration \[FDA\]-approved indication and in the context of a research investigation) * Any concurrent medications that the principal investigator determines could interfere with the trial * Treatment with oral/systemic corticosteroids, with the exception of topical or inhaled steroids or low dose systemic steroids for physiologic replacement (e.g., Prednisone ≤10 mg/day) * Exposure to household contacts =\< 15 months old or household contact with known immunodeficiency * Allergy to measles vaccine or history of severe reaction to prior measles vaccination * History of receiving the measles vaccination with the "killed vaccine" between 1963-1967 without subsequent re-immunization (2 doses) with the active, live vaccination."