Usually serial or VIN numbers are located on the frame toward the front, sometimes on a plate, sometimes stamped in to the frame, sometimes on the inside of the frame rail. Repainting by previous owners may have covered it, look closely. Sometimes there may be a sticker inside a cabinet or storage compartment door.
Procedures and options for getting a new title and/or registering the trailer will vary widely depending on your state. A call to the DMV may help.
A Google search for vintage trailer forums may lead to more information.