There are couple of good choices for your grocery needs on Dauphin Island.
The Ship & Shore is on your right just as you come on the island. They have just about everything the vacationer needs for an extended stay on Dauphin Island - and we do mean everything. If they don't have it, you can probably get by without it.
If you're headed off the island you'll pass right by Greer's Food Tiger at Bayley's Corner.
Further north you start to get back to the city and the civilization you came to Dauphin Island to get away from. There you'll find, among others, the Tillman's Corner Wal Mart Super Center, a Target, a Kmart and just about all the other big box stores you can think of. Tillman's Corner is about a 30-45 minute drive from Dauphin Island.
Finally, during the the growing season there is usually a Farmer's Market in the park at 7am each Saturday. It's worth getting up early for this one, even on a Saturday.