Ronnie's Bakery is not too far from the resort, just down 278 towards Sea Pines Circle. It's kind of tucked away near a gas station, I think. We haven't been there in a number of years, so I can't really speak to the quality. It's probably more of a bike ride, rather than a walk, from the resort though.
Hilton Head Diner also has "to go" bakery, and is within a decent walking distance. Again, haven't been there in years.
BUT, if you truly don't mind getting in the car, go to Signe's. It's a restaurant/bakery and was recently featured on Rachael Ray's "30 Minute Meals" on the Food Network. YUM!! It's also a great place to go for breakfast, but be prepared for a bit of a wait -- well worth it, though.
I don't think there is a Krispey Kreme on the island. The closest Atlanta Bread Company is near the Wal-Mart, which is several miles away.
Have a great time at Hilton Head!!