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.
A Google web search for chomping at the bit returns about twice as many results as a search for champing at the bit. Champing at the bit is still used just slightly more often than chomping at the bit in published books, but you can see chomping rapidly gaining ground in the Google Ngram chart that shows how often words and phrases are used:
Chomping has actually overtaken champing even in published books if you filter the search so that you look at just American English.
”Champ” is an older, more formal word that means to gnash or chew on something (as a horse would a bit), but we don’t use champ much in American English. We are more likely to use the more informal word chomp, which is probably why people remember the phrase incorrectly.
Frequency alone doesn’t make chomping at the bit correct, but most style guides and dictionaries I checked also refrain from going so far as to state that it is incorrect. This is language change at the point where you probably can’t win. No matter which form you choose, some people will think you got it wrong.
Still, if you’re writing for school or for readers who are versed in English, champing at the bit is probably the safer choice.