Your pictures helped a lot less than I thought they would. I can see it is chain drive from the trans back and the spec sheet says it has a CVT. That stands for
constant velocity transmission. Between the engine and the transmission there should be belt drive two pulleys system. They are more than pulleys, they are clutches, as rpm increases one expands while the other contracts changing the gear ratio. You're going to have to help out a little. Remove the clutch cover,
between the engine and the transmission, jack up the back of the kart and support it with some jack stands or something so that the back wheels could turn of they were working. Start the kart and give it a little gas ( no need to rev way up ) and look for what is not spinning. The clutches both on the engine and the driven or secondary as well as the sprockets both from the transmission and the axle are fixed with a key so they don't freewheel.
If it is internal to the transmission I probably won't be able to help, but first let's make sure everything else in the drive train is working.