Can 2 12 volt weed wackers power a go cart with a 200 pound man in it and at what speeds?

Answer 1

I don't think weed wacker motors are suitable for this purpose. These motors are all about speed, not so much about power (even if you attach a gear, as you would have to). Also, with 12 Volts and 2 motors, you will use quite some Amperes to move a 200 Lbs man = you need a big battery.
Unless you are happy with crawl speed.....
There must be a reason why most robot hobbyists use 24 Volt wheelchair motors at $150.- a piece.

Answer 2

You'll have to gear them down some, but they should work. I have no idea how much speed you could get out of them, that depends on the available power and the gearing ratio.

Doug