TIL it's technically "champing" and not "chomping" at the bit Definition of CHAMPING AT THE BITBut as so often happens, as long as enough people use the wrong way of saying something, it becomes accepted as a right way of saying it
champ - Wiktionary, the free dictionary So, champ changed spelling to become chomp? It seems like it from what I've read. If this idiom has been around long enough, then it might have fallen victim to the changes in English phonology.
... and since the word "champ" is ultimately an imitative (sound word? sth like "woosh" or "plonk") then I guess it'd be more sensitive to changes in phonology than other words.