Language evolves over time and that process is often driven by popular usage. This may be one of those instances where popular usage has so far outstripped the original use of a phrase that it is beyond correcting. Nevertheless I persevere if only to note the passing parade of English usage. The original and traditional form, going back at least to the early 16th century (1530s), is “champ” and “champing.” The earlier expression was “champing at the bit.” It refers to impatience. That said, even in the 16th century the verb “to champ” referred to chewing. In 1638 Daniel Featley used it to mean “to chew.” The difference seems to lie in the distinction between its transitive use as distinct from its intransitive use. Used absolutely or intransitively it is “champing at the bit” (1852; Thackeray). A transitive verb passes its action to its object. An intransitive verb does not. In this case the bit is not in a state of having been champed. The action remains with the subject, the one champing. So a book is read (transitive) but a bit is not champed (intransitive).
The expression “chomping at the bit” arose in the US in the 1930s. It’s a small thing to exchange an o for an a. In the modern period, judging by what I hear, most American English speakers seem unaware of the older expression. It may be that using it now may seem pretentious and I am not champing at the bit to bring it back but I hope that readers will not swallow the newer usage without at least being aware of the change.
The historical references are drawn from the respective entries in the Oxford English Dictionary.