I have definitely heard horses belch. I realise horses can't vomit but burping is actually a different mechanism. In humans, the act of vomiting is quite complex and involves a coordination of various internal muscles while I suspect burping is more of a passive process. Therefore, just because horses can't vomit (not sure exactly why, probably due to their anatomy.. somehow) I don't see why they couldn't burp.
When I bestride him, I soar, I am a hawk: he trots the air; the earth sings when he touches it; the basest horn of his hoof is more musical than the pipe of Hermes. ~William Shakespeare