One definition of bit is a metal mouthpiece used for controlling a horse, and one definition of champ is to bite or chew noisily. These are the senses meant in the idiom champing at the bit, which refers to the tendency of some horses to chew on the bit when impatient or eager. In its figurative sense, it means to show impatience while delayed, or just to be eager to start.
The idiom is usually written chomping at the bit, and some people consider this spelling wrong. But chomp can also mean to bite or chew noisily (though chomped things are often eaten, while champed things are not), so chomp at the bit means roughly the same as champ at the bit.
In fact, chomp, which began as a variant of champ, is alive in English while the biting-related sense of champ is dead outside this idiom, so it’s no wonder that chomping at the bit is about 20 times as common as champing at the bit on the web. Champing at the bit can sound funny to people who aren’t familiar with the idiom or the obsolete sense of champ, while most English speakers can infer the meaning of chomping at the bit.
Still, if you’re writing for school or for readers who are versed in English, champing at the bit is probably the safer choice