There are a few things you can do, to set up a kart for him, but the big question is: How fast are his reflexes, reaction time, Is he pretty sharp and aware, quick to react? I know that his legs are not functional as you stated above, but it is extremely important to know his sharpness to be able to judge if he can drive or not.
It is a big deal and a safety issue, you really don’t want him getting hurt or others.
Now moving on to the options hoping that he is capable and able to control the kart and speeds.
There was a video posted here sometime last year, and I’m pretty sure it’s on youtube.com, in fact he posted a few videos, step by step, showing how to build all the linkages for hand controls, he also built a crane to help him pick himself up and drop back inside his kart. He had no use of his legs at all.
Please understand that I don’t have a lot of info about this kid, and I can only give you a few pointers, based on the info you gave us, but if everything looks good, and he can do it, then there are a few options and yes, it is very possible to get him to drive and have fun like a normal kid would.