That's interesting they want you to do that. I guess I don't have any experience with the APHA, but I had a similar "problem" with a horse we bought last year that was AQHA. It took almost a full year until he was finally registered in my name b/c they tried to contact the in-between people ... who they couldn't get a hold of. But they were the ones to handle contacting the previous owners .... not me.
Anyway, good luck! Just follow whatever protocol that APHA says you have to follow. It's really more of a paper trail, to make sure the horse hasn't been stolen somewhere along the way.