If you are not planning on using actual soft-body physics then this effect can be done in vertex shader with a bit of math. Aka find contact points, send them to material, in material pull vertices away from contact point to the center of the wheel hub and push some of them aside. Vertex colors can be used as masking for this - so that only flat side stretches to the side and not contact surface itself.
Otherwise if you really want to keep your setup with a lot of bones - make a small cylinder as a collision boundary for maximum tire deformation and find position of the bones using ray-trace or sphere trace.