I think you're right, Ebony. Most people will just say they're fine and don't need anything even if they do. I think a lot of people just don't want to be a burden to others.
I think groceries are always welcome. Grieving people may not feel like cooking for themselves, so a casserole or other dish they can just heat up might be a nice thing to drop off. If you're visiting someone who's grieving, offering to help with something is a nice gesture, too. I think just knowing that you're there is comforting for a lot of people.