Normally large chains don't welcome dogs, but Bass Pro Shop does. Typically we wouldn't recommend taking them to somewhere like this unless it was really special but the Bass Pro Shop in Memphis is. Why? It's a giant pyramid! The place is absolutely huge with tons of stuff to check out, including live bass and ducks (so be cautious with your dog) plus tons of outdoor dog gear.
In Gatlinburg, there's a fantastic stretch of little specialty shops with a nice walkway that meanders through all of them. Quite a few of the shops are dog-friendly (and this is usually posted outside on signs) and there is even a pet-specific shop you can go in to check out gear, grab a drink, and treats.
From our research, there are at least 50 dog parks scattered throughout the state. Some of them are small and easy-access while others are beautifully landscaped and sprawling like Sevierville. Do yourself a favor and plan a mini dog park tour of places with water features and lots of shade so you can beat the heat and still have fun.