Huge wegoshop.com fan here - we used 'em for the first time this past July, and couldn't be more pleased. There is the aforementioned fee, but seeing them come up to our rooms, coolers and bags in hand, and knowing we didn't have to lift a finger - not bad. As another poster stated, they'll shop wherever you like (I'd steer 'em toward Publix for decent prices).
Disney does charge $5 for their own delivery service. They deliver after 5 PM, and they will store perishables in the refrigerator. Their selection is limited.