Not sure why you ask this in a family friendly show. But i may aswell play along. Horses i think can eat meet, deers eat birds but they don't need to eat it. They are herbivores. Meaning they can live on eating green, unlike carnivores or omnivores where its more required to consume it for vitamins needed in order for your body to function.
If you want to see the disturbing illustration, google "deer eats bird" or "horse eats chicken" you have been warned though. As for it, no. I don't see a reason too. I don't even think they kill animals in MLP world to eat, their world is too perfect and you'd likely be locked up in a dungeon since i can imagine it being against the rules in that world.
Or i found a video explaining it. But its not recommended if you don't wanna think of ponies/horses as meat eaters. But it explains it very well, if the information is accurate i don't know but it doesn't seem unlikely to be untrue. I'll put it in spoilers and you can see if you wish. But generally speaking herbivores eats green all the time, period.