Veterans Services Support Act - Directs the Secretary of Veterans Affairs (VA), during any period after September 30, 2013, for which VA appropriations for FY2014 have not been enacted, to ensure that VA employees who are excepted from furlough provide: (1) veterans service organizations with the same access to VA facilities as such organizations had on September 30, 2013, (2) veterans with the same access to veterans service organizations using such facilities as was provided on September 30, 2013, and (3) such services as necessary to ensure the safety and security of such veterans and veterans service organizations in the same manner as was provided on September 30, 2013.