"Inoculate" is a much older word (from horticulture), and the practice is older as well, since it dates back to the 10th century in China and the Middle East. It involved the deliberate introduction of the same virus from a pustule on a smallpox victim into the body of another. It conferred immunity by the same organism.
Figuratively, it means to defuse or remove a threat or criticism by discussing it first. ("Defense counsel inoculated his witness against mention of his criminal record on cross-examination by bringing it up in his direct examination.")
"Vaccinate" is not used in this way. It originally involved the use of a related but not identical virus ("vaccinia" is cowpox) to induce immunity, so the idea of reducing a threat by mentioning it first doesn't fly.
And "contretemps" originally meant a fencer accidentally running his opponent through in practice because he was "out of time" i.e. out of synch. In modern usage, it means an unfortunate accident (from 1802) or a quarrel (from 1961).
"Chomping" is not preferred but is not technically incorrect, because it means the same thing - chewing noisily, which is what the original expression means. From a horsey point of view, though, "champing at the bit" shouldn't mean that the horse is anxious or excited. We want our horses to mouth the bit and move it around, because it means that their jaw and poll are relaxed and they are receptive to the bit. I even use a copper center on my horse's bit to encourage salivation and chewing.
He does misuse "literally" all the doggone time, though.
You know what's going to happen if he or his people read this?
He's going to deliberately misuse "literally" about ten times as much --watch.
That has happened to him before, here, but with a different word, and things here went nuts.
Actually it was clear from his reaction that he was simply giving us a playful shout-out.
Eh...but yeah, probably the "literally" one is the only one that reallly bugs me.
I’ve written to him at least a million times and told him NOT to exaggerate; too!