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.
Is it chomping on the bit or champing at the bit?
Curzan says using either three or four syllables is acceptable. Here’s another question: If you are impatient or eager to do something, we use a metaphor with a horse, and the horse is doing something with the bit. It’s known as “champing at the bit,” but many people say “chomping at the bit.”