The idiomatic expression that indicates impatience stems from what a horse does to its bit, the metal piece of a tack that crosses the horse’s mouth. Champ here means to chew or bite at without the intention of swallowing. The idiomatic expression is now commonly written as chomping at the bit, which comes from the American English dialectal variant, chomp, meaning the same as champ. The variant styling is now so common that it’s moot to argue against it, though champ may still serve those who are careful with their language.
Bryan A. Garner suggests that the two words have undergone differentiation, with champ retaining its original sense and chomp adopting a meaning akin to chewing noisily with the intention of eating, though no other usage writer mentions such a difference. It is also likely that modern writers use chomp for both biting noisily and scarfing down food.