Levee Vegetation Review Act of 2013 - Directs the Secretary of the Army to undertake a comprehensive review of the Corps of Engineers policy guidelines on vegetation management for levees in order to determine whether current federal policy is appropriate for all regions of the United States. Requires the Secretary to examine the guidelines in view of factors including: (1) the varied interests and responsibilities in managing flood risks; (2) preserving, protecting, and enhancing natural resources; (3) protecting the rights of Native Americans pursuant to treaties and statutes; and (4) factors that promote and allow for consideration of potential variances from national guidelines on a regional or watershed basis.

Directs Corps of Engineers Regional Integration Teams representing districts, divisions, and headquarters, in consultation with state and federal resources agencies and with participation by local agencies, to recommend to the Chief of Engineers vegetation management policies for levees that conform with state and federal laws and other applicable requirements.

Directs the Secretary to: (1) solicit and consider the views of the National Academy of Engineering on the engineering, environmental, and institutional considerations underlying the guidelines; and (2) revise the guidelines based on the results of the review and report to Congress. Requires the revised guidelines to: (1) provide a practical process for approving regional or watershed variances from the national guidelines, and (2) be incorporated into a proposed manual.