They still had plenty, but the AT&T store didn't. I kept asking the AT&T guy if I could go back to the Apple Store to complete the process, but he seemed puzzled and reacted almost like I had suggested going to the moon to finish. He insisted on ordering one for me, and the order still hasn't even shown up in AT&T's order status system.
The activation systems definitely are recovering though, so at least my old iPhone works again.
Edit: Whoops... I finally found the order. The key was to enter the store's ZIP code and not mine.