As far as I know, there is no way to register without knowing the parents and having the proper paperwork. The DNA that registries have on file allows the listed parentage to be checked against what is on file for that horse and make sure a new foal can be from the listed parents, but last I knew, registries do not have searchable databases of DNA that can pick unknown parentage out of their entire list, or anything like that. Calling the registry would be the best bet to make sure what they can and can't do, but I doubt the horse would be registerable with nothing at all to go on.
If you COULD find out the names of the horse's parents, I believe you would still have to get signatures of the parents' owners on the registration application-- but again, checking with the registry is always the final word. Here's their contact info-- Welcome to Arabian Horse Association - Contact Us
I would call, as an email might not be returned in time for you to know before the sale.