Definitely take tea bags. THe supermarket you refer to is Goodings- now Winn Dixie and it is not in walking distance because of the roads not having pedestrian walkways. Disney have a way of closeting you during your stay with them, to keep out the competition
You can always take a bus to one of the other Hotels if they do not have what you want. I have stayed at the Wilderness Lodge a few times and their shop is stocked well with essentials and goodies. They had small coffee jars [folgers I think] and milk, cereal etc