I would imagine with enough skill and ingenuity you could put a dollar bill acceptor on anything. You would need to get a bill head (Mars is most common), buy or build wiring harness(es), wire it up to 110v AC (not a problem, as you have that in the cabinet already), mount it in the cabinet (not too big a deal if you're ok with woodwork), and tie the pulse line into the coin switch to give the game credits.
My question to you is why would you want to do that? Out Run, Continental Circuit, etc. is OLD school...why would I put a dollar in it? Don't get me wrong, I love my video games as much as the next guy, but the average consumer has a hard enough time putting physical cash (i.e. bills) into a NEW video, never mind something that's 25 years old. Plus, am I going to have time to play 4 games? Or are you charging premium pricing (50 cents) per game?
If you have any other options for equipment, may I suggest an upright Megatouch? Comes with a bill acceptor on it, and the option of having 50 +/- games usually is a godsend for the location. Plus, it's a smaller footprint, easier to move, and most people don't mind throwing a bill into a Megatouch. I think it's more subconcious, as you're used to doing it in the bars (or wherever) already. As an added bonus as well, there are usually games for any age group there, as well as multi-player games.