I wouldn't put a dog in a shopping cart (why?) but people take service dogs into grocery stores because they need them.
Having said that, non-service dogs (just pets) are allowed in a lot of places here. Costco, Home Depot, Tractor Supply, Ace Hardware, coffee shops, bookstores, grocery stores, fabric/craft stores, bars, restaurants.
Keep in mind that for most of the year here it's too hot to leave a dog in the car and too hot to walk them outdoors. People bring dogs with them into stores and walk them in air-conditioned malls.
As long as the dog is leashed and well-behaved, I don't mind.