First let me say that we usually have our own car when we are at WDW, however, I have seen many posts that say their limo driver will take them to get groceries. Where are you thinking of faxing your order ahead to? If it's the resort...I probably wouldn't because I'm sure the prices are marked up and you won't get as much for your money.
There is a brand new Publix supermarket right off SR535 (from I-4) that is really nice and clean and has reasonable prices. The Publix is in the back of the Prime Outlet mall and Mary Queen of the Universe Shrine off on SR535. I'm sure your limo driver will know exactly where it is. Hope this helps.