When:
During Part 1, when you are exploring Mêlée Island.
Method:
Without giving away too much, you'll reach a point where you're able to "borrow" a cookbook from the governor's mansion. Take this to the chef in the back room of the Scumm Bar, and whilst he's distracted reading it, pick up the mop from the back room. Attempt to steal the mop by leaving the Scumm Bar.