Hey Ludsky,
I haven't tested it, because I have not currently installed the STK Blender add-ons, but I believe you have an incorrect pivot point on the wheels.
The pivot point indicates the center of a 3D object. You notice the transform arrows are below the actual 3D mesh if you select a wheel of the kart. That's where your current pivot point is. The wheel will rotate around this point, so let's bring it to the center of the mesh.
In the tools panel [T] there is an option "Set Origin", which can set the pivot/origin point. Select "Origin To Geometry" [Shift-Ctrl-Alt-C] for centering it according to the geometry.
Another thing I noticed is the object transform of the wheels. You can see the object is 90 degree rotated in the euler x-axis (marked in orange). I don't know if this is supposed to be, but I could imagine that it can lead to problems. You can fix that without rotating the actual mesh by applying the transform from object space into mesh space. Here's how you'd do that:
Select the wheel in the 3D view and hit [Space], to open up the search input. Type in "apply" and you'll find an option called "Apply Object Transform", click that.
This will open up an options menu in the in the tools panel. There you can select what transform should be applied. Select "Rotation".
Your wheel should now have a correct position of the pivot and a correct transform. Hope that helps.
Let me know how if you have further problems.
Cheers
Julian