I don't think you want to use a starter motor. They are designed to be used at 12 volts and are only supposed to be used for a few seconds at a time. You might get away with a treadmill motor, or your best bet would be to hit up craigslist for a cheap electric scooter that the batteries have died on. That way you can salvage the motor, controller, throttle, power switch, sprocket, and if your lucky some batteries.
Brushless motors are AC induction motors that have 3 wires coming out of them and are much more efficient (read alot less heat that fries motors) than your common DC motor. Efficiency is important because things run cooler and more power gets to the pavement. Since you guys are running go-carts that are not used for transportation, you can actually get away with running 2 of the Bosch 36 volt toolpack batteries with 30 minute charger for $150 shipped. These will put out 10-15 minutes of high amperage fun followed by a 30 minute charge, get more and rotate out for constant fun. If you want to be cheap and run for 30 minutes get Lead Acid, but charge time will be more like 8 hours

You will only save like $20 dollars anyway.