It's not any different than going to Walmart yourself and getting groceries -- they just put them in plastic grocery bags. I've never had a problem with anything refrigerated or frozen. (Well, I've never ordered ice cream.)
The key is to be home when they deliver it; so, you can put those things away immediately. They offer two-hour delivery windows and usually deliver within the window. In my experience -- I'm disabled and use them for groceries almost weekly -- about 5% fall just outside the delivery window. (Usually 10-20 minutes late.)