Lots of luck finding pet-friendly in Gulf Shores . . . you can pretty much rule out any condo, and none of the big 4 or 5 rental companies have anything pet-friendly.
I would suggest you try Dauphin Island instead. Dauphin Island is the next island to the west. It is connected by a bridge to the mainland, and you can also take a car ferry across the mouth of Mobile Bay over to Gulf Shores.
Here is a link to a list of some pet-friendly properties on Dauphin Island: