Is it possible? Yes....but is it practical? Probably not. Once you start looking at trailers under the 1 grand mark you are starting to see bad structure, flooring issues, along with the wiring, tires, and rusting. There are deals out there though and some people might be looking to sell to get some quick cash but for the low end of things I would look to spend $1500 - $2500 on a older but usable trailer.
If it is a wood floor those aren't too hard to replace and either is the wiring... but once you get into rusting through that gets spendy. I had a 15 year-old stock trailer that was starting to rust and I looked into have it sand blasted and painted but it was going to cost a little less then what the trailer was overall worth. While I know that cost is a concern don't only go for that... make sure you horse/horses will be safe everytime you take them for a ride. Good luck in your search!!!
I would recommend looking at Horse Trailer World- Used trailers for sale, New trailers for sale, trailer classified ads, trailer manufacturer news and more.