Sunitinib Malate, Paclitaxel, Doxorubicin Hydrochloride, and Cyclophosphamide Before Surgery in Treating Patients With Stage IIB-IIIC Breast Cancer

Inclusion Criteria: * Be informed of the investigational nature of the study and all pertinent aspects of the trial and must sign and give written consent in accordance with institutional and federal guidelines * Have a histologically-confirmed diagnosis of breast cancer that is locally advanced or inflammatory; inflammatory breast cancer is defined as erythema and peau d'orange involving half or more of the breast with a histologic diagnosis of breast cancer; the finding of focal dermal lymphatic involvement on histology does not constitute inflammatory breast cancer * Have selected stage IIB (T3, N0, M0) or IIIA (T3, N1-2, M0 or T0-2, N2, M0) disease judged primarily unresectable by an experienced breast surgeon or otherwise deemed appropriate candidates for neoadjuvant treatment or stage IIIB (T4, any N, M0) or stage IIIC (any T, N3, M0) disease * Patients must have a performance status of 0-2 by Zubrod criteria * Absolute neutrophil count (ANC) \>= 1,500 cells/mm\^3 * Platelet count \>= 100,000 cells/mm\^3 * Serum creatinine =\< 1.5 x institutional upper limit of normal (IULN) * Bilirubin =\< 2.0 * Serum glutamic oxaloacetic transaminase (SGOT)/serum glutamic pyruvic transaminase (SGPT)/alkaline phosphatase =\< 2.0 x IULN * Have a multi gated acquisition scan (MUGA) or echocardiogram scan performed within 3 months prior to enrollment and have a left ventricular ejection fraction (LVEF) % greater than the institutional lower limit of normal * Be willing and able to comply with scheduled visits, treatment plan, laboratory tests and other trial procedures Exclusion Criteria: * Have evidence of distant metastases * Have tumors that overexpress human epidermal growth factor receptor 2 (HER2)/neu as evidenced by 3+ staining by immunohistochemistry or gene amplification by fluorescent in situ hybridization (FISH) * Have received any prior chemotherapy or hormonal therapy for breast cancer * Have received prior radiation therapy or prior definitive surgery for breast cancer * Have a clinical diagnosis of congestive heart failure or angina pectoris or any of the following within the 6 months prior to study drug administration:, myocardial infarction, coronary/peripheral artery bypass graft, cerebrovascular accident or transient ischemic attack, or pulmonary embolism * Have ongoing cardiac dysrhythmias of National Cancer Institute (NCI) Common Toxicity Criteria for Adverse Events (CTCAE) version 3.0 grade \>= 2 * Have uncontrolled hypertension (\>150/100 mm Hg despite optimal medical therapy) * Have pre-existing thyroid abnormality with thyroid function that cannot be maintained in the normal range with medication * Have a known, active infection * Have any prior malignancy except for adequately treated basal cell or squamous cell skin cancer, any in situ cancer, adequately treated stage I or II cancer from which the patient is currently in complete remission or any other cancer from which the patient has been disease-free for 5 years * Human immunodeficiency virus (HIV) positive * Are receiving or planning to receive any concurrent anticancer therapy while receiving protocol treatment * Are receiving or planning to receive concurrent treatment on another clinical trial (supportive care trials or non-treatment trials, e.g. quality of life (QOL) are allowed; participation in the companion imaging trial, dynamic contrast enhanced-magnetic resonance imaging (DCE-MRI) and fludeoxyglucose F 18 positron emission tomography (FDG PET) with Kinetic Analysis to Monitor Breast Cancer Response to Neoadjuvant Sunitinib and Metronomic Chemotherapy is also allowed) * Be pregnant or breast feeding; female subjects must be surgically sterile or be postmenopausal, or must agree to use effective contraception during the period of therapy; all female subjects with reproductive potential must have a negative pregnancy test (serum or urine) prior to enrollment; male subjects must be surgically sterile or must agree to use effective contraception during the period of therapy; the definition of effective contraception will be based on the judgment of the principal investigator or a designated associate * Have other severe acute or chronic medical or psychiatric condition or laboratory abnormality that may increase the risk associated with study participation or study drug administration, or may interfere with the interpretation of study results, and in the judgment of the investigator would make the subject inappropriate for entry into this study