Edit: just read your post update.
Hm.. either way, I doubt they'll allow it to happen. Phone plans are strongly tied to credit (plus I wouldn't respect a store very much if they did not ask for identification with card purchases) and it's inevitable that they'll check.
What's your rush though? The first time iPhone was released there were no shortage issues. Go back another day with your dad.
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Oh yeah,
Since iPhone 3G requires in-store activation, it'll prompt a credit check. Either way you'll need someone with credit there to buy the phone and activate the plan.
Apple won't sell the phone without activation, and activation is impossible without credit.
If you and your dad don't have AT&T service, then they'll need to run a credit check to create a new service plan.
If you and your dad are already under AT&T, any modification to the account, such as phone purchases and changes, requires him to be there (since you don't have a credit card I'm guessing that he would be the one that your wireless plan would be under).