Horses are very strongly programmed - even more than humans - to avoid a repeat of a situation which proved to be dangerous.
For example, many years ago I knew a horse named Anglo Starsky in the UK. He was a multi Stakes winning Standardbred gelding, a pacer. The only reason they didn't keep him intact was because he was cryptorchid (which can be genetic).
In his last race he tripped, fell, and ended up underneath and tangled with the sulky. He was not seriously hurt...but he would never go between the shafts of a vehicle again. He was retrained as a saddle horse and had a good life, but if he so much as saw any kind of horse drawn vehicle, he would freak out to a dangerous degree.
PTSD in humans results in flashbacks to the original event triggered by something that reminds them of it. Starsky would melt down when he saw a cart or carriage because it reminded him of the original accident. Was it PTSD? It's hard to tell - we can't ask a horse if he's having flashbacks - but a horseman can tell if a horse's mind is "with" them. Dissociation may not be something we can ask the animal about, but it's something an experienced horseman can sense.
So, can horses get PTSD? I lean towards yes...but it's not something we can, as of right now, prove one way or the other.