Unless there's a flat tire or something, taking the DME bus will take LESS time than waiting for your bags at baggage claim AND making a grocery stop with a car service.
There are good reasons to prefer a car service to DME (personalized service, private ride), but getting to your resort quicker isn't one of them, ESPECIALLY if you plan on making a grocery stop.
Also, just a friendly observation, not angry or yelling or anything

... the "free" grocery stop ends up being more expensive than the premiums Disney puts on their groceries at the resort gift shops, because the "free" grocery stop is a perk of the not-free car service ride, which could run about $120 round trip including tips. I wouldn't pay for a car service ONLY in order to save a few bucks on some groceries.
I used Gooding's (
www.goodings.com) twice for grocery deliveries and was fully satisfied each time. And even though I've never personally used Garden Grocer (
www.gardengrocer.com), I've spoken with the owner, Brian, and he seems to be a VERY conscientious and nice guy, and I plan to use his service in the future.
And as others have mentioned, if you're looking just for bottled water and maybe some non-perishable snacks, go online to
www.staples.com and sign up for their Staples Rewards card -- that's all you need to do to qualify for free shipping, and you can have them ship it to your resort. WDW resorts will hold shipped packages for guests for up to 5 days in advance of arrival.