You're going to need more than batteries to run a shop vac. 12A @ 120V is 120A @ 12V. Assuming perfect conversion, which ain't gonna happen, you'll need roughly two of my car, running, to deliver the amps.
Nothing that will plug into the lighter will have much power. Those circuits are built for 150-200W, which is about enough to run a good coffee maker. One without a built-in grinder. Pushing the amps you'll need to power a vacuum of that size will require wiring more like your starter and less like your iPod charger.
Silly question, but had you considered a broom?