Yes, if you call AQHA with the horse's registered name, they will tell you the name and last known address of the most recent owner they have on record for the horse. You can then contact that owner to get a signed transfer, and the registration certificate, if they have it. If they no longer have the certificate, you need to have them send you not only a signed transfer, but also a signed/notarized affidavit for a duplicate registration certificate.
Here are the forms you will (might) need-- http://www.aqha.com/forms/pdf/Transf..._02_01_Web.pdf http://www.aqha.com/forms/pdf/AffidavitDupCertWEB.pdf
Here is the phone number for AQHA--
Customer Service --(806) 376-4811
The owners of record do not HAVE to do this for you, but hopefully they will. Sometimes there is a reason that papers were not passed along-- payments weren't made, agreements not completed, etc. Like Kevin said, the last recorded owner has the right to refuse transferring for whatever reason, but hopefully you can find them and they will help you.