Hi, Alanna! The closest Publix supermarket to Downtown Disney is about four miles away. In the heat of the Florida sun in September, this is probably not going to be a very comfortable walking distance. If you won't have a rental car, you can get a taxi at your resort to make the trip.
There is a Goodings Supermarket just at the end of the Hotel Plaza Boulevard, and is much easier to walk to (although keep in mind, it is going to be a hot walk there and back, so keep that in mind if you intend to buy frozen foods or perishables. The Goodings market has a great selection of grocery items, but you'll probably find they are more expensive in general than what you are likely to pay for things at home. The difference in the cost of the groceries may make it worth it to pay for the taxi to the Publix.
Also in the area is a Winn-Dixie Supermarket. They are a little bit closer than Publix, but for most people - myself included - definitely not within walking distance. The prices at the Winn-Dixie are probably the best of any of the three grocery stores, but your best buys require their rewards card. You can apply for the card online at the Winn-Dixie website.
I hope that helps! Have a wonderful vacation!