The proper way to say it is champing at the bit.
To champ is to do a repetitious, powerful opening and closing of the mouth that produces sounds when the teeth hit together. Also means to show impatience at being held back or delayed.
For the non-horse oriented crowd a bit is a metal piece that is placed in the horse’s mouth and attached to the reins. It helps to control the horse when it is being ridden.
Champ turned into chomp like stamp become stomp, if you know what I mean.
As time goes on chomping at the bit is becoming more widely accepted.