It depends on the weight of the trailer when loaded, and if your truck has a tow package an what it's rating is. You'll have to check with Ford on the specific truck you're looking at - if you go to their website with a VIN number, you can see if a truck has a tow package. DO NOT trust a dealer, check with Ford.
It probably is not a good idea. Most people would advise to go straight to a 3/4 ton trailer for towing. An F250 in this case.
You have to check around and see trailers similar to what you'd want to buy, what they weigh unloaded then try to figure out what they might weigh when loaded with 2 horses and tack. Then pad that number up a little bit. THEN when you are looking at trucks, compare that number against what the truck is rated to tow. Some versions of the F150 are only slated for 5000 pounds, which if you get a steel trailer (heavy!) isn't going to leave you much wiggle room, and most people prefer to tow well under their limit.
Hope that all makes sense!
(When I started looking around earlier in the year, I read a lot and got advice from people on a few boards and ended up buying an F250. I got a super great deal because I was able to buy a truck that had aftermarket mods which made it illegal in a neighboring state where I bought it, but were mostly legal in mine. For under $100 I was able to make my truck street legal.)