There are many, many older horse trailers like this on the roads here in Ohio. In virtually all of these cases, the trailers did not come with a title like they do now.
In Ohio, to get license plates for one of these, you have to have the trailer weighed by the State Patrol and get an official "weight ticket".
With a bill of sale and the weight ticket, you can get plates each year. Have you checked with your state registrar as to the rules? Even if you find out the year and make, I don't know how you could prove it.