Goats can't vomit. Real burping as we do is rare. He is probably bringing up his cud. If he were getting gs he would bloat. Bloat is readily apparent since the stomach swells up large and hard and sounds hollow like a drum when tapped. Then you should call your vet asap. Keep an eye n yiur babies and you will see that once he "burps" he starts chewing again or cudding.
Goats are a 4 chambered stomach species. Their food goes into several chambers to be digested and extract all possible nutrients from it. They are very efficient. I am assuming these goats are off milk. The way the rumen works is that it allows the goat to eat, then they regurgitate their half digested hay into their mouths so they can ches it again and get more nutrients out. then it goes through the rest of the digestion process. This process makes it possible for them to eat and digest very fibrous foodstuffs like dry hay forage, leaves twigs, etc.
Watch adult goats, cows and sheep, and you will notice that they eat for a while, then lay down somewhere and continue chewing until they get up and eat again. They fill their rumens, then regurgitate and cud - that is the chewing they are doing while resting. After sted food they keep bringing up, their rumens will pas it along to the rest of the intestinal tract and they will be hungry again and resume grazing, foraging, or screaming for you to bring them more feed pronto! LOL
By the way, try not to have your face next to his when he brings up his cud. It can be stinky! We have all been caught by cud breath!