The Peerless 700 series is the trans most people run in the racing mowers and such and have been fitted into lawn karts by people wanting speed selection, reverse, and mostly - to develop their fabrication skills. They have been used widely in small rear engine and stand behind mowers and have many variations and gear ratio selections. Properly built, they can be reasonably sturdy, but they do not like being shifted while moving. Also, racing karts remove the (internal) chain and disable reverse, as when the chain fails, it destroys the case. The only way to use one in a kart and use a horizontal shaft engine is to also run a 90* gearbox, which adds expense and fabrication - as has been said - they are designed to be used with a vertical shaft engine.
As they are typically mounted - the shift shaft is on top. Power goes into the bottom of the unit, usually through a v-belt pulley, and the belt is also the clutch through a pedal/idler setup. Not all models have outputs on both sides. On the one shown, either side can be fitted with a chain drive output sprocket, and many times there is a small brake on the other end. Rotate mounting it 180* and the output shaft changes direction.
They do not like shock loads, as in clutch dumps. They do not like being shifted while moving - you can, but shouldn't; it damages the drive keys. They tolerate torque better than RPM, so it is better to add speed with your final drive sprocket ratio than by spinning the input faster. And you still have to use a 90* adapter to run one with a Clone motor, which, might be $150-200 new or a $20 garage sale steal.
So the point is - can it be done? Yup. Easy? Nope? Cost effective? Not even close. LOL
With the proper teacher it would be a great learning experience, because it would challenge
and develop your fab skills. More of a project to use up a bunch of parts laying around,
than one where you have to source them and probably pay retail.
I hope you enjoy whatever project you end up building. Willie