What are their shuttles to WDW like?
Limited schedule, for a fee (except nightly cash rentals, I think), with (I believe) 24 hour advance reservations. Call the resort directly for up-to-date info. It's a very short drive, so using a car is advisable if possible.
Where is the closest non-Goodings grocery store?
5 years ago, we got on I-4 west, exit at Rte. 192 west in Kissimmee, 1-2 miles on the right was a Publix with a very good selection and reasonable prices. There may be something newer and closer now.
And are there any nearby restaurants that deliver there?
The Orlando area has a delivery service for most nicer restaurants, with a service fee above whatever you order. I don't remember the name of the service. There are some prepared foods available at the onsite poolside restaurant near the "superpool." I remember there being a card in our room advertizing delivery from a pizza shop but that was 5 years ago. HTH.