Imagine you have a perfectly dense incompressible fundamental particle.
Strike that particle with another of the same type.
Will they vibrate?

If something is not compressible. then I don't see how there could be any sort of shockwave that would cause the particle to vibrate.
Am I correct in this assumption?

If not, please explain why.

I think you're correct.

It probably won't vibrate internally but if that particle is in a field with which it can interact (and is stable within that field) then it will vibrate about its former position while dissipating the energy.

If it were perfectly dense and incompressible, would there be energy to dissipate, or would it simply pass right through it undeterred as a sort of kinetic superconductor?
Would it even absorb any energy at all without compression?
What would be the mechanics of transferring, absorbing and dissipating any kinetic energy?

It wouldn't store energy but it will gain kinetic energy.