Do you have a title
from the previous owner in Washington? If you do and a VIN is listed on it then get it stamped on the tongue at a garage or shop. Stamp sets to do this are available at places like Tool King or Harbor Freight for very reasonable prices so you might be able to get a set and do it yourself.
Most older trailers had the VIN stamped on the top or side of the tongue so check closely in that area. If the tongue has been painted several times the number may be hidden under the many layers of paint
. Also check under the propane
tank mounting base if your rig has one. I've found a few hiding under there.
Most states and provinces have a procedure to assign a VIN to homemade trailers so you could also research that path. You might need a bill of sale for the "pieces" you bought to build that homemade trailer though.
I'm currently going through a similar situation to get a title
for a vehicle I bought from a junkyard. The title was deleted from state records when the vehicle was scrapped out. I just needed the body but, of course, that's where the VIN is stamped.
It might take a while and many trips to the motor vehicle office but it can be resolved.