There are a few things to clarify:
Thermal radiation is EM radiation, generated by the thermal motion of particles in matter.
All matter with temperature above absolute zero emits this kind of radiation, and the motion of particles will create:
Even the CMB is thermal radiation.
It is very important to understand that thermal radiation is a form of converting kinetic energy into EM radiation.
Thermal energy consists of the kinetic energy of random movements of atoms and molecules in matter. All matter with a temperature by definition is composed of particles which have kinetic energy, and which interact with each other. These atoms and molecules are composed of charged particles, i.e., protons and electrons, and kinetic interactions among matter particles result in charge-acceleration and dipole-oscillation. This results in the electrodynamic generation of coupled electric and magnetic fields, resulting in the emission of photons, radiating energy away from the body through its surface boundary.
Now you are saying that you still want to differentiate this from thermal radiation, and you are not talking about thermal radiation. You say that you are talking about EM neutral objects, you say like atoms and molecules, and if they vibrate, do they emit EM radiation or not.
Now you are assuming that you can have a atom or molecule isolated in free space, vibrating.
The definition of vibration is mechanical oscillation.
You are talking about composite objects, like atoms and molecules. So I will assume you are asking about EM neutral composite objects, that are isolated in space, vibrating (mechanically oscillating).
Now atoms are defined to be EM neutral, but there exists dipole and quadrupole, and higher order fields, and these fields can be attractive, and this is why we can talk about the QM phenomenon of covalent bonds, this is how atoms can combine into molecules, sharing valence electrons shells.
You are distinguishing atoms in these bonds in solids, liquids, and gases, having degrees of freedom of rotational, translational and vibrational motions (interpreted as temperature, or heat capacity), from the free neutral atom in your case, that vibrates in free space.
You are saying, that a neutral object, in free space, isolated, would vibrate, and thus create EM radiation. Now as per SR, an accelerated charge will emit EM radiation.
In your case, since we take the atom as EM neutral, though it is vibrating, it will not emit EM radiation, as long as:
it is isolated, free in space, not in a molecule, or in a lattice, or in a liquid or gas
its EM charge distribution is in equally distributed (dipole moment)