As long as the stallion has AQHA papers, and does not have the notation on his papers about having excessive white, he is an acceptable parent for an ApHC eligible foal.
If he is not already, the stallion needs to be "listed" with ApHC before his foals will be registerable, which means sending in a front-back copy of his papers, the listing form, the fee, and a waiver form for DNA if his DNA is already on file with AQHA (the waiver form allows ApHC to get his DNA type from AQHA.) If he is not already DNA typed, that needs to be done.
Then before November 30 of the breeding year, the stallion owner needs to file a stallion breeding report with ApHC listing any ApHC mares he bred, breeding dates, etc. It can be filed late, but it costs more.
Your ApHC mare also has to have DNA on file before her ApHC foal is registerable.
Regarding "back in the day" getting papers yanked if Appaloosas bred paints-- I think it is an old wive's tale. I have copies of the Appaloosa News from the 1960's on, and ApHC rule books from the 70's on, and have tried to make a point of studying the various rules, changes, the whens and why's etc-- and have never come across any such rule. There were articles and pamphlets discouraging that type of cross, because the foal would not be registerable-- but not any rules or penalties against it I can find.