Chip Foose had one of those old bumper car bodies sitting outside his office door a couple years back. When I asked him about it he indicated that he wanted to mount it on a modified Quad chassis. He did a couple quick sketches and it looked really cool, but his sketches as I remember had more of a slammed look.
If you Google "street legal bumper cars" all of the hits seem to indicate that they are using 750cc bike engines in these. Now, with the exception of the silver car that has Show Tubes on it, the rest seem to have single Quad type exhaust. Regardless I think that wheelie bars would be mandatory.
With the high CG and the live rear axle I'm sure their a handful to drive.
In Florida it seems to me that if it has 4 wheels and you have to straddle it, it can't be licensed. So much for using a Quad.