First off welcome to the forum!
Now to the go kart questions.
You say you have lots of old riding mower parts! It really depends on your abilities. You could build one from scratch, that gets pricey but is also rewarding. Most guys on here use the Harbor Freight predators, they run $100 and make 6.5hp which is enough depending on frame size and that kinda thing.
If you have a trans axle from a mower and an engine that's vertical shafted you basically have an entire drive-train right there! So, do you plan on building from scratch or buying a frame and restoring?
Again, WELCOME to the forums!