Accountability for Veterans Act of 2013 - Requires that, whenever the head of the Department of Defense (DOD), the Social Security Administration, or the National Archives and Records Administration (agency head) receives a request from the Secretary of Veterans Affairs (VA) for information necessary to adjudicate a veterans' benefit claim, the agency head shall, within 30 days, either: (1) furnish such information; or (2) submit an explanation for not being able to furnish such information, as well as an estimate of when the agency head will be able to furnish such information. Provides for the continuous resubmission of, and required response to, requests not acted upon within the original or estimated period. Requires the Secretary to notify the appropriate claimant of each request made and explanation and estimate received.

Directs the Secretary to: (1) keep records of the information requested and the time it takes to receive responses, and (2) submit semiannually to the congressional veterans committees a summary of such records.