Photos of an Idler Clutch
These images show how a crude clutch can be created by using an idler (or sometimes called, a tensioner) pulley wheel. When the idler pulley is away from the belt, the belt is very loose and no drive will occur. When you want the transmission to engage, you move this 3rd pulley wheel in against the drive belt, causing it to tension and then to grip the pulley wheel on the engine and rear axle.
This clutch system is used on the lawnmower powered wooden gokart as on: http://www.kartbuilding.net/Lawnmower_Powered_Wooden_Go-Kart_Plans/belt-tensioner-clutch.html
Just for information, I came across this very old lawnmower in an old building in Toronto! It goes to show you what you would find when you keep your eyes open. The pulleys, and belt would be perfect for use on the wooden go-kart.