Gearbox
If you are lucky enough to have a "Tractor Lawnmower Gearbox" complete then you are sorted. For this you do not need the original engine off the tractor lawnmower . You will be using your lawnmower engine to drive this gearbox. However it is a very good idea if you could just buy this part (gearbox) off a scrapyard or someone who has broken down tractor lawnmowers. You don't want the engine which may have been the part broken in the first place. All you want is the gearbox which is in every case I've seen is seperate of the engine and you shouldn't have to pay too much.
The great advantage is that these gearboxes are meant to take an engine with a verticle drive shaft. Also you will have the nearly all essential gears for your kart. The clutch may also be included and that's another bonus. Therefore it's not a bad idea to spend a day looking for one!.
Usually the original engine with the verticle drive shaft drives the gearbox via a drive belt and pulleys. However this belt and pulley system is in line and thus has very little heating and slippage problems to worry about. There may be a clutch system built in here, in that the clutch is worked by putting a tension onto the drive belt which will drive the pulley wheel on the gearbox.
If you are buying the "tractor lawnmower" gearbox then try and get the linkage belts and pulley wheels, as this will save a lot of messing around.
So on your kart you have the engine fixed at a height to make the drive belt reach the gearbox. You have the gearbox fixed to the chassis/ kart. From this gearbox it's a matter of connecting up the output shaft to the rear axle. This in my opinion is best done using two sprockets and a chain. With this system you can speed up the gearing to make your kart go faster than a tractor lawnmower !!. Again I have info on chains and sprockets on the main Drive page previous to this one.
If it's a big enough gearbox it may have the rear axles comming out of it instead of one drive shaft. In this case your best option is to make these stub axles bigger and to put wheels on either end. To do the gearing up of the drive you can step up the diameter of the pulley on the lawnmower engine shaft with respect to the pulley on the input shaft of the gearbox.
It all depends on the type of "tractor lawnmower gerabox" you get your hands on!.
This is a neat and compact tractor/ ride-on mower that would be suitable. The engine is on it's own at the front and the gearbox/ differential is contaned in the rear axle driven by a long drive belt from the engine. Hiowever this is brand new and would cost around £1333.
Here are more contraptions from which you will get a gearbox/ differential suitable for your kart.
Look around, you may be suprised what you will find and be able to use.!!