Before 1981, VIN's were not standardized at 17 numbers and letters. You are probably looking for a shorter combination with a system used by the manufacturer. If stamped into the tongue, it may be covered with multiple layers of paint and you may have to remove the paint. Look first on the driver's side of the tongue and then look elsewhere. As long as you have a title with a VIN and the trailer is not moved to another state, licensing is easy. But if the trailer goes to another state in a sale, it will have to have something on the trailer for a VIN inspection in that state. You may be able to apply for a new VIN in that case, but in some states it can be very difficult and expensive.
There are a number of websites that offer search for a specific VIN. I don't know if they are free or not. They may not have a VIN for a specific trailer. When we sold our trailer, the buyer checked out the VIN on one of these websites and couldn't find it. So, the website he used was not complete. Perhaps none of them are, but I don't know. You can search the VIN and see if it matches the trailer. Airstream may have VIN information too. I know you are looking for a VIN on the trailer, but confirming the trailer and title match is a good idea. Mistakes are made and the title, registration or a VIN on the trailer may not match. I had a car like that and it was so long ago I can't remember what I had to do, but I think it was fairly simple.
Hopefully you find the plate and a stamp VIN on the tongue, they are readable and they match the title.
The Airstream is sold; a 2016 Nash 24M replaced it.