IMO it's a little heavy for grocery shopping, but I guess it could sit in the carts seat!
You know, I have no idea if grocery stores have free wifi, but I kinda don't think most would. You can call and ask them. More places are getting it, but a big store would require a lot of bandwidth which is a lot of money.
Type in "free wifi" with the name of your city and while that probably isn't a complete list it's a start. Looks like mostly coffee shops, hotels, a few fast food, Barnes and Noble...
I just tried for mine and came up with a list, but there were no grocery stores on it. I might try different cities nearby, too.