TAYLOR: You would kick Tiny Tim’s crutch out from under him, wouldn’t you?
LUKE: If he asks for a free cup of coffee, gimpy’s going down.
Timothy “Tiny Tim” Cratchit, a character from the 1843 novella, A Christmas Carol, by Charles Dickens. In the story, Tim is the young son of Ebenezer Scrooge’s employee, Bob Cratchit, and a very sick boy who needs crutches to walk. Scrooge is shown that Tim will die in the future unless he receives medical help that Bob cannot afford on the salary he receives from Scrooge. This is one of several visions which cause Scrooge to reform, and the story states that Tiny Tim didn’t die, and that Scrooge became a second father to him (presumably paying for medical treatment).
Gimpy is a derogatory name for someone who walks with a limp. The slang dates to the 1920s, and may be a combination of gammy and limp, gammy being used to describe a bad leg.
The Annotated Gilmore Girls 