Yes, all Target stores in Chicago accept the Link EBT card. This includes the Target stores that don't have expanded grocery departments. They have been accepting the card since 2009.
Target as well as Sam's Club, Costco, CVS, and 7-Eleven started accepting the Link card in 2009 because more Americans started qualifying for food stamps during this time.
The use of food stamps reached a high this past September 2012. There was an increase of almost 300,000 recipients in the month of September. Almost 48 million Americans are on food stamps, with Texas, California, and Florida having the most food stamp recipients.