Grocery will probably be busy today AND tomorrow... quite a few people do their cooking the day before, it seems. Your best bet to avoid the rush is to go as early as you can, or, if it's a 24/7 store, as late - even if they're cleaned out, if you can go at night, they've usually started restocking by then, and the crowds are smaller. :) I go late to avoid crowds.
I have to go tonite after DH gets home. I have all my stuff but need more milk and eggs and have to get some aluminum pans and paper plates and cups. Best get some plastic cutlery too. Hmmmm...I'm making a master shopping list for next year once this is all over.
Definitely go later then 9pm or it will be over crowded (used to be cashier for a grocery store and did work the week of turkey day holy hell busy!)...expect the store to be out of certain items...good luck!