I would say a food tourist is someone who travels recreationally for the purpose of dining. Furthermore, the unromantic sounding phrase 'food tourist' rather than something like 'culinary tourist' seems to imply a lack of refinement, so I'm going to make a distinction. Whereas 'culinary tourism' might imply traveling to exotic destinations for exotic fare, food tourism seems to imply domestic travel only.

A food tourist might take a road trip to go to every Waffle House in a certain state, or fly into the Twin Cities in order to eat at the largest food court in America. Maybe even a special trip to Chicago or New York City to hit up Morton's, Peter Luger's and all the other run-of-the-mill 'meat, potatos, and cigar' type places.