We know that most things vibrate at the particle level.
Why is this?
Most would say Thermal Energy.

Thermal energy is not the reason.
Thermal energy is just light.

Kinetic energy?
Kinetic energy is based on mass and velocity.
Should the particle vibrate back and forth, it would violate the law of conservation of energy.

So how do we explain this phenomenon that has been commonly accepted until now?...

Everything moves. I just finished a tpic with someone else stating similar problems. In turn, if you freeze something down to absolute zero temperature, motion still occurs. This is highlighted in the third law of thermodynamics.
Another way to see this, is by seeing what we are left with in the vacuum if remove every bit of matter/energy - you are paradoxically left with an enormous amount of energy - this is the zero-point field, and here everything is bubbling... this is called quantum foam. Here, the uncertainty principle rules the game. Just remember that everything at the lowest levels are jitterbugging, even at ground states of entropy.
NeoNo.1

Given that at at lowest levels everything is jitterbugging is there any way to use variations to either send or receive a signal.

For instance a proton would be locked in an electro magnetic field due to its electron but if you could suspend a neutron in a plasma and if it vibrated differently could that difference be used as a signal transmitter or receiver?

The idea is that we are looking for gravity waves, what if all we have to do is see if space shakes particles or more to the point if something is already shaking space.

If we where able to do this, it would be a very weak transmittor. The more power transmittors, or computational devices come out of using superpositioning, where a value isn;t either 0 or 1, but rather 0, 0.50 and 1. Thus, if we had an atom in a state of superpositioning, a strong electromagnetic field would pull the electrically charged atom out of its superpositioning, and a defined spin would be made for it.
NeoNo.1