You should be doubly thankful that some stores remain open on Thanksgiving. For one, it's a lifesaver when you realize that you invited 15 people over for dinner but own approximately two cups. Second, a last-minute errand is the perfect excuse to get out of a house filled with 15 people who after five explanations still have no idea what you do for a living.
For most of your real shopping, you should wait for Black Friday -- and look at the early online deals from Amazon and Walmart, and make sure to do some shopping that'll get you free Whoppers -- but if you do need to get an entire dinnerware set last minute, or if you somehow forgot to buy a (fashionably small) turkey, there are stores that will have you covered.
Just make sure to be extra nice to all the employees stuck working on the holiday. Oh, and it couldn't hurt to call your local supermarket or big box store before you head out in case they're closed for the day after all.