Have you wire brushed the area on the tongue wher the vin lives? It is usually painted over and will reappear. Do you have the vin on any paperwork? AZ doesn't require the trailer to be inspected to register it, but they do need the vin to check it isn't reported stolen. If you can't come up with the vin the tx code below outlines a procedure to have it inspected and ownership documented, then they will issue a new vin to attach.

TX Administrative Code DMV

Excerpt from Code on VIN

Vehicle identification number. If no vehicle identification number is die-stamped by the manufacturer on a motor vehicle, house trailer, trailer, semitrailer, frame, or item of equipment required to be titled, or if the vehicle identification number assigned and die-stamped by the manufacturer has been lost, removed, or obliterated, the department will on proper application, presentation of evidence of ownership, and presentation of evidence of a law enforcement physical inspection, assign a vehicle identification number to the motor vehicle, trailer, frame, or equipment. The manufacturer's vehicle identification number or the assigned vehicle identification number will be used by the department as the major identification of the motor vehicle or trailer in the issuance of a title.

It does state that tx registration must occur within 30 days of transfer, so if it has gone beyond that I'm sure they'll assess some penalty.