Horses aren’t physically capable of vomiting.
An extremely sick horse (usually with something stuck in its throat) may be seen with a small amount of stomach contents coming out of its nostrils, but they cannot empty their stomachs the way humans can. This is why colic in horses is very serious.
Horses have a band of muscle around the esophagus as it enters the stomach. This band operates in horses much as in humans: as a one-way valve. Food freely passes down the esophagus into the stomach as the valve relaxes but the valve squeezes down the opening and cuts off the passage for food going back up.
Horses, however, differ from us because their valve really works. Humans can vomit. Horses almost physically can’t because of the power of the cut-off valve muscle. Also, the esophagus meets the stomach at an angle which enhances the cut-off function when the horse’s stomach is bloated with food or gas. Then the stomach wall pushes against the valve, closing the esophagus even more completely from the stomach. Normally, the mechanics are such that the horse’s stomach ruptures before the valve yields.
If material does pass from stomach out the esophagus, the horse is dead or nearly so. That’s why horses can’t vomit.