Leuprolide Acetate or Goserelin Acetate With or Without Vismodegib Followed by Surgery in Treating Patients With Locally Advanced Prostate Cancer

Inclusion Criteria: * Patients must have histologic proof of prostatic adenocarcinoma via a minimum of 6 core biopsy samples * Clinical stage T1c or T2 with high-grade disease (Gleason's 8-10) on initial biopsy and prostate specific antigen (PSA) \> 10 ng/ml, or clinical stage T2b-T2c with Gleason's grade \>= 7 * No evidence of metastatic disease as determined by imaging * Initial therapy with antiandrogen treatment is allowed but must be within 4 weeks prior to study enrollment * Appropriate surgical candidate for radical prostatectomy and an Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Absence of major co-morbidity as determined by the treating physician * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>=100,000/mcL * Total bilirubin within normal institutional limits * Aspartate aminotransferase (AST/serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT/serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X institutional upper limit of normal * Creatinine within normal institutional limits OR creatinine clearance \>= 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Patients must have prothrombin time (PT), partial thromboplastin time (PTT) and fibrinogen levels within institutional normal limits and no history of substantial non-iatrogenic bleeding diathesis * Men and their female partners must agree to use two forms of contraception (i.e., barrier contraception and one other method of contraception) during study treatment and for at least 12 months post-treatment * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Histologic variants in the primary tumor (histologic variants other than adenocarcinoma) * Patients who have had chemotherapy or radiotherapy for prostate cancer prior to entering the study * Patients who have received prior treatment with GDC-0449 * Patients may not be receiving any other investigational agents * Patients receiving previous androgen ablation or current androgen ablation of greater than 4 week's duration * Patients who are not appropriate surgical candidates for radical prostatectomy based on the evaluation of co-existent medical diseases and competing causes of death (such as but not limited to, unstable angina, myocardial infarction within the previous 6 months, or use of ongoing maintenance therapy for life-threatening ventricular arrhythmia, uncontrolled hypertension) * History of allergic reactions attributed to compounds of similar chemical or biologic composition to GDC-0449 or LHRH analogues * Patients taking medications with narrow therapeutic indices that are metabolized by cytochrome P450 (CYP450), including warfarin sodium (Coumadin®) are ineligible * Patients with malabsorption syndrome or other condition that would interfere with intestinal absorption; patients must be able to swallow capsules * Patients with clinically important (in the opinion of the treating physician) history of liver disease, including viral or other hepatitis or cirrhosis are ineligible * Patients with uncontrolled hypocalcemia, hypomagnesemia, hyponatremia or hypokalemia defined as less than the lower limit of normal for the institution, despite adequate electrolyte supplementation are excluded from this study * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible * Patients with prior malignancy if there is an increased chance (\>= 30%) of relapse in the following five years (in the opinion of the treating physician) * Patients who have received systemic treatment for cancer within the last 6 months