I can report now that I got my order to go through (space grey, 6, AT&T, 64GB) on the iOS app. I did immediately get a confirmation email as well as an order number, unlike the limbo I'm seeing this year and saw in past years from carrier sites that took a while to catch up.
There do seem to be problems even with the app. It took a number of tries for the continue button to light up and there were a few timeouts after that, most notably on the upgrade check, but all eventually went through. I did not add any accessories, and was offered a choice of shipping or pickup at the closest store. I changed the pickup point to the other local store and the app didn't blink at that.
My husband and I have a shared plan, but I am the master account holder, so it was my SSN that was required (last four), so if you're secondary on someone else's plan, get that info now before you try to check out. I do know that the app does ping AT&T for plan status as it correctly determined that we have a grandfathered unlimited data plan. The confirmation email does state that that plan remains in effect.
As of a few minutes ago, the app was displaying "Available on 9/19", so I'll be picking it up next Friday. Again, this has to be done by the master account holder, so kids on parents' accounts, your parent will need to be the one presenting ID at pickup.
We plan for me to use this phone, since he's not as anxious as I am to play with new toys (the new camera, for instance - I'm the shutterbug, and the NFC feature doesn't interest him) so I'll be swapping the SIM out of my 5s before activating it, but we will need to visit an AT&T store to get him a new SIM as his aging 4s (with fading battery life to match, hence the upgrade badgering from him) has a micro-SIM, which will not fit the 5 and up. Hopefully there will be no problems; we did something similar to this some years ago with a different iPhone release.
If the continue button doesn't light up, change colours and see if that makes a difference, then back to the colour you prefer. And I wouldn't advise switching away from the in-progress order, especially if you are upgrading -- it's not fun to have the carrier's system think you ordered an upgraded phone while Apple's thinks you didn't. Just tap retry when prompted, or press Continue again.