Champ and Chomp are two totally different words. Champ is short for champion, one who is the best. Chomp is to bite. Chomping at the bit is the correct phrase. It refers to a horse biting the bit, the metal part of the reins that is put inside the horse's mouth to aid in steering the horse.

Chomp. "champ" is not a verb. It's a noun. It means "champion".

Both are acceptable. http://grammarist.com/usage/champing-chomping-at-the-bit/

In the U.S., the use of 'champ' will almost always mean 'champion.' Chomp is far more common, and chomp isn't used much in the U.S. either. However, champ is more vigorous chewing than chomp, but their definitions are quite similar. In the U.S "Chomping at the bit" is more common than "Champing at the bit.