Hi! I know for a fact that my delivery people offer almost every time to bring the groceries right into my house. As for if they'll put the refrigerated items away, I don't know for sure, but I'd encourage you to message Shipt on Facebook and ask them. Their customer service on Facebook is awesome, and they get back usually within two hours.
I would be surprised if they said their drivers won't do that, but I am curious to know what you find out!