Do you have the mare's registration papers, or just a copy?
If you have the originals and the breeder's signature is on the transfer/sale line, you can register your mare in your name.
If you don't and the breeder is still the owner of register, you need to contact them, not the 'in between' owner.
Once the mare is registered in your name, you can register the foal as a Half-Arabian as long as his dam is a purebred. If she's only a partbred, the foal is ineligible to be registered with the AHA.
You don't need the stallion owner's signature in order to register a partbred foal, as long as the mare is a registered purebred.