I took a dog into a big supermarket beginning with T. I did it because I reckoned it was not safe to leave a dog tied up outside. If it became afraid it could bite. Anything could happen. With me, the dog was under control.
Needless to say I was told off. I did something legal but against company policy.
I think these places should have a policy for people who have their dogs with them. It would appear that this is not the case. I am looking into the legal aspects of who is responsible if the dog is left outside, which appears to be a fundamentally bad idea... The carpark is not publically owned land and it is the decision of the landowners to disallow other animals inside.
I think that if shops have a No Dog policy, they should be absolutely clear about where you can leave a dog safely.