Nope, just turn back now before you dig yourself into a deep sinkhole of vertical shaft ****.
I'm going to answer 99% of the questions you probably have:
Why can't I use a vertical engine?
Because no go kart parts were designed for a vertical engine, and making them is time consuming and expensive. Worm drives don't work, you can't just twist a belt, and a proper 90 degree gearbox might just cost you as much as an entire motor.
Can't I make a vertical engine horizontal?
That's a pain in the *** and the amount of time and money spent converting it, you could have driven to harbor freight and gotten a Predator.
What about the transaxle in the lawnmower?
Over 10mph it becomes a ticking timebomb. They are designed for lawnmowing, and not much more. Its already been long since known that they aren't any good for go karts.
What about putting a Predator/other horizontal shaft engine into the John Deere?
Carsandcameras did it....but thats still more expensive than just buying an old frame and putting a new motor on it, and likely more expensive than just getting a premade go kart used. Also lawnmower frames aren't generally the stiffest given they're used as a lawnmower not a 35mph go kart. Oh yeah and Ike nearly died on their go kart mower when it flipped.
If you have any other questions, feel free to ask but I would seriously advise you to consider another path to a go kart.