Yes, if you want your ownership of her recorded with ApHC and your name to appear on her papers, you have to have a signed transfer form from current recorded owner. (whoever's name is currently on her papers.)
If you show in open classes at ApHC shows either you would need to be her recorded owner and a current member, , or her recorded owner would have to be a member. If you want to show youth or non-pro (not sure of your age) either you or an immediate family member have to be her recorded owner.
If you are showing somewhere other than ApHC sows, it should not matter, as open shows do not require registration papers. However if you want to enroll in ApHC's ACAAP program (ApHC will give recognition/points for showing your Appaloosas in open al-breed shows) I think you would need to be a member and her recorded owner-- or, at the least, her current recorded owner would have to be a member.
If you call the ApHC (208-882-5578) they will give you the last known address of the mare's recorded owner-- it may or may not be current info, but its at least a start.