A lot of the retail stores will allow you in as long as they don't serve food (health code violation.)
Generally, the stores don't really allow you in with a dog (most stores have some kind of rule against this) but they'd rather see you shop and spend money then send you away. The whole "we have rules but we turn a blind eye" kind of thing. They're more concerned about an animal making a mess then they are about you having a dog that is not bring attention to itself.
If you don't call attention to yourself (especially if you're using something that doesn't scream that you're carrying a dog) you should generally be fine. Most employees and managers want to avoid having any kind of confrontation that would potentially cause a problem.
I've worked part-time in several retail stores over the years, and in all the stores I've worked at, the general consensus has always been:
"as long as the feet don't touch the floor leave them alone"
Your best bet is to really just... risk it (and don't test on days when you really have to shop there
). Attitude makes a huge difference too, since you're a lot more likely to be noticed and checked out if you appear nervous and such. If they make a big deal, just quietly leave and put em on your "no" list.