The Answer
The original phrase is “champing at the bit”, but “chomping at the bit” is considered acceptable according to many sources.
According to Merriam-Webster champing/chomping at the bit means “waiting in an impatient way to do something”. You’ve probably heard both versions of the idiom and you may have wondered which is correct.
The original idiom was champing at the bit. The bit in question is the bar that connects to a horse’s bridle and sits in its mouth, and champing means (according to dictionary.com) “to bite upon or grind, especially impatiently”. When a horse is impatient it will champ on its bit, often in a nervous manner.
Unfortunately, our crack team of researchers (actually, just me and Google) were unable to find the exact time period when this phrase emerged, but we did find an unverified reference here that suggests it was the mid-1600s. Merriam-Webster dates the verb champ back to the 14th century.
It does seem that the alternate form of the idiom chomping at the bit arose in America in the first half of the 20th century.
The original phrase is, indeed, champing at the bit, but chomping at the bit emerged in America in the 1930s according to the Oxford English Dictionary and chomp has overtaken champ in common use.
Grammarist makes the case that chomping at the bit is now an acceptable use of the idiom. “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.”
Effectively the argument is that we should accept an incorrect or misunderstood idiom once it reaches a certain critical mass of usage. In this case, many sources, including Merriam-Webster and NPR agree that both variations are considered acceptable.
You’ll have to decide for yourself if you want to use the correct version or the common version.