The "wheels" are really the hubs, mostly from the various Speed Champions cars among others.
The inside of the wheel takes a standard stud, so you should be able to use:
Plate Modified 2x2 with Pin Hole
Technic Pin 1/2
Wheel Cover 5 Spoke
Beyond the primary question of how to build these particular wheels, there are a few common themes among wheels and ...
If you want to stick to your current design (despite its obvious flaws), then here's another idea. Let's go!
As you can see, I tried to rebuild the crucial part of your framework.
Alright. Now two things I've noticed. You're using axles with stop. And there's some play between the bush and the wheel.
Okay, firstly on axles with stop.
Here's two pairs ...
I think the real problem is the design and the way you used the motors.
If you motorize only one of the wheels with the medium motor to get the other one to move, then the one that's closer to the motor may go faster and you'll probably end up with a slight deviation to the left, which also includes the other wheel snapping off.
Also, using the large servo ...
You can attach couple of half bushes or full bushes on inner side of the axle near Technic beam like on the other side you have done with a gear. If axle is still trying to slide - fill in the entire inner axle part with bushes.
Here is picture of suggested fix. Here you can see extra half bushes added to fix the axle the same way gear fix the other axle ...