A Cheesy or Gaudy, pompous, or arrogant person who also happens to have cheeks that look like they're stuffed with shit (similar to the way chipmunks store food in their cheeks). The term is very similar to Ass Face but can be used to refer to a cheap and gaudy event, show, or presentation frequented by Chipmunk Cheese types.
See Rush Limbaugh
Eddy: "So what did you think of my boss Monty?"
Suzy: "Not much. He's such a chipmunk cheese!"
"The 2008 Republican convention was just so much chipmunk cheese!"