Getting a delivery is SO MUCH BETTER than having to take a taxi and then shop yourself. I've done the "shop for yourself" thing during non-Disney vacations, so I've seen both sides, and believe me, getting a delivery is absolute heaven compared to doing it yourself.
There are 3 delivery options ... I'm familiar with only 2 of them.
I've used Gooding's twice, and I was totally satisfied each time. I've spoken with the owner of the local Garden Grocer franchise, and he impressed me a lot in terms of his committment to quality service to his customers.
Gooding's is a local supermarket that also offers delivery. Garden Grocer is a stand-alone grocery delivery service.
I'm about to explain a grocery delivery method. It's the way Gooding's does it, and it's one way that Garden Grocer can do it. You set a date of delivery but not a time. They don't even attempt to deliver it to you at your room (you might not have even arrived in Florida yet!) . They deliver directly to your resort's Bell Services folks. Bell Services then calls your room (if you've checked in yet). If you're in, they say they'll be right up with the groceries. If you're not in, they store your grocery order for you and even keep the cold stuff cold (which is bagged separately and labeled); they leave a message on your room phone.
When it's convenient for you, you call Bell Services from your room and ask for them to bring your grocery delivery to you. A tip would be appropriate.
As far as the ordering process itself, I can only speak to Gooding's method, as I've never actually used Garden Grocer. Their website has a pretty small selection, but you can order anything that they stock on their shelves, except alcohol. Before your trip, you give them your credit card information, but they won't charge your card until just before the delivery ... there would be no way for them to know the final price before that, just in case they are out of a certain item, or if you ordered deli or produce that must be weighed, etc.
Garden Grocer CAN deliver alcohol.
The $20 Gooding's fee and the $12 Garden Grocer fee are much cheaper than round-trip taxi fare plus tip, and it takes you no time out of your vacation!