Though I can't find any proof of this, I would assume that they are too proud to be carnivorous. However, I also can't find anything about them eating any herbs or fruit from the Forbidden Forest.
Does anyone know what they eat?
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It’s never mentioned, either in the seven books or Fantastic Beasts and Where to Find Them, what centaurs eat. The only way we can narrow the possibilities down is that centaurs live in forests, so it’d have to be something (whether plant, animal, or both) that exists in the forests they live in.
“The centaur is forest-dwelling. Centaurs are believed to have originated in Greece, though there are now centaur communities in many parts of Europe. Wizarding authorities in each of the countries where centaurs are found have allocated areas where the centaurs will not be troubled by Muggles; however, centaurs stand in little need of wizard protection, having their own means of hiding from humans.”
- Fantastic Beasts and Where to Find Them
However, there’s nothing that implies whether they’re carnivores, herbivores, or omnivores. Their lifestyle could indicate either. They respect nature, which could imply that they’re herbivores, but there are many cultures that respect nature but also eat meat. They’re skilled at archery, which could imply they hunt for food, but it could also be a method of defense - when we see them use their archery skills it’s in combat or to protect themselves from something they perceive as a threat.
“That is because it is a monstrous thing, to slay a unicorn,” said Firenze. “Only one who has nothing to lose, and everything to gain, would commit such a crime. The blood of a unicorn will keep you alive, even if you are an inch from death, but at a terrible price. You have slain something pure and defenseless to save yourself, and you will have but a half-life, a cursed life, from the moment the blood touches your lips.”
This quote from a centaur (especially the part that I bolded) can easily imply that they would look askance at killing animals for food. If that is the case then they are presumably herbivores.
Elsewhere, the centaurs confirm that it is not only unicorns that they don't kill, rather they don't kill anything innocent:
“Nevertheless,” said Magorian calmly, “the slaughter of foals is a terrible crime... We do not touch the innocent.