I don’t have a car and take a rolling insulated bag with me to the market. (I think I got mine at Home Goods or TJ Maxx(. If you have to do a big stock up, consider doing delivery. Also, some of the markets do have customer parking but I’ll defer to @Knowsstuff about the markets in Wrigleyville.
In terms of food - again knows stuff will have good recommendations but super close to where you are staying, we like Smoke Daddy (and they are beautifully decorated for the holidays) and Mordecai but check to make sure things are going to be open on NY day.