No stores for an ESA (or emotional support animal). To be able to go into stores your dog needs to be VERY well-mannered. No greeting people, eating food off the ground, barking, etc. They need to be rather invisible until you need them. Only Service dogs can be in public stores.

With an ESA, you just get to rent anywhere, even places with no-pet contracts and don't need to pay a pet deposit, fly in cabin, but there are some restrictions, and that is it. It doesn't have to be a dog, it can be a cat, small horse, bird, or even reptile!