DM: Alright, I think I'm all set up and ready to begin. Is everyone else good to go?
Twilight Sparkle: As prepared as I'll ever be.
Applejack: Let's get this trainwreck a-rollin'!
Rarity: Practically chomping at the bit, I daresay.
Rainbow Dash: This final boss has gone undefeated for too long!
Fluttershy: Let's just all have a good time, okay...?
Discord GM: You know me! I'm always ready, able, and willing to take to the stage!
Pinkie Pie: Huh? Should I clear the snacks off the table, then?
Huh. Wait, are these human players in a pony game, or pony players in a pony game again? (It's the first, isn't it?)
'cause I kinda wonder about 'champing at the bit' as a turn of phrase in Equestria. Maybe it's meant about money-bits? 'cause I can only imagine that a 'bit' in Equestria (as refers to the piece of equestrian equipment) is closer to like, a Grill or a ball gag in social use. But then, they use saddles as corset-analogues, so...
I think it might be a phrase that originated in certain militaries. I can see siege equipment having bits that can be bitten and used for leverage rather than forcing a soldier to put on a full harness to aim or reposition a weapon and then take it off before they fire. There may also have been helmet mounted weapons that would include a mouth bit the soldier could bite for extra grip or control.
It is the former. But since Newbiespud is choosing not to create pictures of the human players, their speech bubbles get attached to their ponies, even when speaking OOC. So Rarity's player is using a human expression (with a horse pun) without implying it exists in Equestria.
Although it still could; fashion bridles appear in the show without bits, but in Bridle Gossip, Appletini outfits Rainbow Crash with a bridle with a bit, so the concept is at least known.
In fiction, it's often more important to write things to be believable rather than factually correct. Most people think it's "chomping", so it's entirely plausible that Rarity would also think so.