The horse must meet the height criteria, be inspected by an AMHA official, get DNA'd (twice to make sure the horse is not being substituted for another) and also makes sure the horse is not a dwarf. The pedigree may or may not be unknown, however when hardshipping into AMHA, the parents are listed as unknown even when you know who they are and maybe they are registered R or something else. This is the last year to hardship anything. Horses born in 2011 and beyond will no longer be able to be hardshipped- that is closing.
If you have an unregistered horse, an R horse or Shetland, that would meet the height requirements, etc... to qualify for AMHA, you can have them hardshipped into the registry so they can be AMHA registered too. They have to be three years old to be able to do so. It used to be five but recently the rules changed to three years old to go permanent.
I have a 31.75" mare that came from AMHA stock, but her dam went oversized, so was only eligable to be R. I hardshipped this mare back into A this year so her foals could be registered A again also.