A relative moved out of her house into an Airbnb and will be moving to another one in month or so. ID is still under the last address (same state but a good distance away) because you can't use that type of temporary rental for a permanent address. Can she legally buy a gun from an FFL/store before moving into her new place? She tried recently and they said pretty much "no you can't but come back when your ID is up to date".

If she changes her DMV address to the new apt, then that's new new address. Whether it's 1,2,3,4... months it seems to me irrelevant.
All rentals are temporary addresses if you don't own it. I would think a more important question would be... does the Airbnb owner allow firearms in the apt.

Federal law requires that the buyer provide their current residence address on the Form 4473.
If their government issued photo ID does not show that address, then additional government issued documents showing the buyers name and current address may be used. It's in the instructions for Questions 10 and 26.a on the ATF Form 4473.

"no you can't but come back when your ID is up to date"  isn't federal law, but that dealers store policy.

OK, thanks. I thought it was something like that. Kinda caught me by surprise when she texted me what happened.