We always use Stuarts Cove, and we book through a different excursion company rather than with Disney because it is less expensive, and because they have an awesome two tank shark dive. Stuarts Cove does have the ability to look up dive card information if forgotton, but if Disney was there verifying it without the Stuarts Cove computers, that would mean they wouldn't be able to look it up. They are extremely strict about certification.
I've done their diving with only a resort certification as well, and it was great...it was a two tank, one shipwreck, and one shark, although to be honest I think they were stretching it by letting me do the shark dive, because 1) it was a LOT of sharks and 2) it was too deep for the resort certification. I had my own dive master and my husband is a very confident diver, so basically I had the two of them looking out for me.
I agree with the other poster, it would be better to get certified first, that way you don't have the restriction. We just did the shark dive two weeks ago...my husband LOVED it. I didn't dive, I have some ear issues, but the video is really amazing.