In fact yesterday we took Nana (our fully trained dog) and Bell (our Service dog in training) to the grocery store. Bell has no harness or vest and I was asked about it and all I had to do was explain that she was in training (not all states provide access laws for dogs in training but most places are quite leinient as long as the dog is not disruptive) and we had no trouble.
The local Lowe's home improvement store lets dogs in for no good reason.
I've take 'Pookie The Abused One' in there for desensitization/socialization sessions.
She did really well....:)
[really gotta go now or Mom won't get anything for her birthday tomorrow!]...;))