ACCESS (ADA Compliance for Customer Entry to Stores and Services) Act of 2013 - Amends the Americans with Disabilities Act of 1990 to prohibit an aggrieved person from commencing a civil action for discrimination based on the failure to remove a structural barrier to entry into an existing public accommodation unless the owner or operator of such accommodation: (1) is provided a written notice specific enough to identify such barrier; and (2) has, within specified time periods, either failed to provide the aggrieved person with a written description outlining improvements that will be made to remove such barrier or provided such description and failed to remove such barrier.