This randomized phase I/II trial studies giving leuprolide acetate or goserelin acetate together with or without vismodegib followed by surgery to see how well they work in treating patients with prostate cancer that has spread from where it started to nearby tissue or lymph nodes. Androgens can cause the growth of prostate cancer cells. Antihormone therapy, such as leuprolide acetate or goserelin acetate, may lessen the amount of androgens made by the body. Vismodegib may slow the growth of tumor cells. Giving antihormone therapy together with vismodegib may be an effective treatment for prostate cancer.

PRIMARY OBJECTIVES:

I. To assess the difference in less than or equal to 5% tumor involvement between patients between the two arms.

SECONDARY OBJECTIVES:

I. To assess differences in hedgehog signaling, androgen signaling, markers linked to high grade prostate cancer (PCa) progression, proliferation, apoptosis, and the expression of androgen producing enzymes in the tumor microenvironment between the two arms.

II. To assess safety of preoperative GDC-0449 (vismodegib) in combination with luteinizing hormone-releasing hormone (LHRH).

III. To assess the difference in proportion of patients with negative disease surgical margins between the two arms.

IV. To collect and archive tissue from the primary tumor, bone marrow and blood (serum, plasma), bone marrow aspirate for future study.

V. To assess difference in relapse rate (biochemical, objective) and time to progression.

OUTLINE: Patients are randomized to 1 of 2 treatment arms.

ARM I (androgen-ablation therapy and vismodegib): Patients receive LHRH analogue comprising leuprolide acetate intramuscularly (IM) or goserelin acetate subcutaneously (SC) on day 1 and vismodegib orally (PO) once daily (QD) on days 1-28. Treatment repeats every 28 days for up to 16 weeks in the absence of disease progression or unacceptable toxicity.

ARM II (androgen-ablation therapy): Patients receive LHRH analogue comprising leuprolide acetate or goserelin acetate as in Arm I. Treatment repeats every 28 days for up to 16 weeks in the absence of disease progression or unacceptable toxicity.

After completion of study therapy, patients undergo radical prostatectomy.

After completion of study therapy, patients are followed up every 6 months for up to 8 years.

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