His debut episode is currently the only time he's ever been involved with the main plot of an episode.
He is rarely seen without the mayor. Even when Chompy's not seen in person, he's usually mentioned only by the Mayor.
In Dream Goat!, Chompy the Goat was somehow able to give Vicky a wedgie (one that was bad enough to make her go "Eep!") despite not having hands/fingers/thumbs--at the time, Vicky was shown wearing a pair of bright pink underwear.
It's widely believed that the reason why we never see his family is that the mayor ate them post-Vicky Gets Fired. But this theory has yet to be verified.