Avila Beach is a really small community, so you won't find any major grocers in this area. There are a few small mom and pop stores, though. I've heard wonderful things about Avila Valley Market & Deli along the Bob Jones bike trail near Sycamore Springs. The owners will be re-opening Avila Grocery & Deli in the Avila village and will, I'm sure, have it open by the time of your visit.
The Custom House, Fat Cats, Olde Port Inn, The Gardens of Avila and a few vendor carts near the beach and pier are your closest dining options.
Custom House serves breakfast-lunch-dinner.
There are more options in nearby Pismo Beach:
Del's Pizzeria (Italian dishes, too)
Old Vienna (German)
Splash Cafe (famous for clam chowder served in a sourdough bread bowl)