Newer trailers had a sticker applied that had the VIN and a few other tidbits. If that's gone or faded, you'd have to talk to the trailer manufacturer to see if there is any recovery of the info.
Depending on your state, you may have to go the home-made route which is getting tougher to do. Mine is now requiring receipts for the materials used in the construction or you're sol.
Of course as mentioned, if you have the paperwork for the trailer, it should be on it...
