Cat

Where did the word cat get the synonym vomit? All I can find is the brit slang to Shoot the cat. Any clues?

Cats yack up hairballs all the time. Maybe that's a clue.

CAT - to vomit. 1785 Grose "Vulgar Tongue: Cat or Shoot the Cat." To vomit from drunkenness. 1966 Longstreet & Godoff "Wm. Kite: 138: He knew I was going to be ill, cat my breakfast.
From Random House Historical Dictionary of American Slang, Volume 1, A-G by J.E. Lighter, Random House, New York, 1994.