We usually stay near SC and our modus operandi once we've cleared the airport is:
Check out the villa and take photos before bringing in the cases;
Drive to Publix about 1.5 miles west on the 192;
Return to villa and make our own sub rolls;
Chill in the jacuzzi with a glass or three of wine.
There is a supermarket at the Formosa Gardens Shopping Center - it's changed hands several times, but I think it's now operated by Wynn Dixie - it just isn't as good as Publix in my opinion. Actually, the newer Publix at Kyng's Heath Road (east on the 192 where it intersects with Poinciana Boulevard) is better. It's not too far away, but probably further than you'd want to drive when you've just arrived.
We always buy fillings for our own subs - beef, chicken, pepperoni, dill pickles, various cheeses, mayo, mustard, tomatoes, lettuce. Publix also has a counter where you can order them freshly made if you prefer (together with hot chicken, etc). The first time we stayed in a villa we did this. The subs come in 'wholes' or 'halves' and our friend decided to order a whole one 'with everything on it' (whilst in Rome and all that). DON'T DO IT! We could hardly stand for laughing - we were literally in tears. The sub was the size of a loaf and the filling was about 6 inches thick!