You will need a go-kart frame that is capable of housing the size and weight of your engine. Most 5 to 7 horsepower engines will fit a standard size kart. The vertical shaft engine you choose can be from a riding mower or a push mower, either will work for your go-kart. Larger engines in the 13 to 22 horsepower range, which you can find on riding mowers, may destroy the transmission or kart. You will want one that can support both your vertical shaft engine and lawnmower gearbox comfortably. Your gearbox should will need to have non-trans axle set-up. This will give you a better sense of the dimensions that will be comfortable for your frame. Most frames have a width that varies between 1/3 and 2/3's the wheelbase of your go-kart, with wider models giving better stability.

Summary:
Measure the size of your engine. Pick a suitable go-kart frame. Draw out a true to size basic design of your frame with chalk.