You've got several ideas here, let me try to break them up for you.
Imagine an electron not moving existing in empty space. How is it attached to the EM field?
A better way to think of it is that the EM field exists, the electron does not. Rather, EM fields are substantial, real things that carry energy and momentum, and electrons are merely a QM manifestation of them.
What does the EM field appear like surrounding it?
You can study up on EM. I made a series on YouTube covering Griffith's Introduction to EM.
The short story: Your stationary electron would create an EM field pointing inward. The strength of the field would be proportional to 1/r2, r being the distance from the electron to the point in the field.
Electrons can never be not moving, electrons intrinsically vibrate, for...no...reason?
You're talking QM here. The fundamental issue is the Heisenberg relationship. Because the momentum of a particle is, roughly, the Fourier Transform of the position of a particle, if you know where it is, it could have any momentum, and if you know its momentum, it could have any position. By "position" and "momentum" I mean the probability distributions of them.
Imagine an electron intrinsically vibrating in empty space. How is it attached to the EM field?
Same answer as above. The EM field is the real deal, the electron merely a manifestation of it.
What does the EM field appear like surrounding it?
A moving current (stream of charged particle with constant velocity and direction) creates a magnetic field that wraps around it. This magnetic field acts on other currents by exerting a force perpendicular to the current and the magnetic field. It's very strange stuff indeed.
When you take a moving particle, the situation becomes considerably more complicated. See, you have to take into account what happens when charges increase or decrease, or currents increase or decrease, and that brings in all kinds of wonderful things. The long and the short of it is that every time a particle accelerates, it creates a EM wave, AKA, light or radio. A passing charged particles leaves a "wake" in the EM field as well.
A vibrating charged particle will create a steady signal, a radio wave or light wave or gamma wave, depending on how fast it is oscillating.
An electron alone in empty space would vibrate...how and why?
I don't know where you get this idea. If you had an empty universe except for a single electron, it would be perfectly still. I mean, depending on how you look at it, taking SR into account.
The vibrations I think you are referring to are what happens when you put an electron near a proton. The result is that the particles interact such that they form a hydrogen atom. No radiation is emitted, surprisingly, because of QM. The key fact here is the distance between the electron and proton merely change the results of the equation. Your hydrogen atom could span the size of the entire universe, if you put the electron far enough away from the proton. Over time, the atom would radiate as the electron moved closer to the proton, just like excited hydrogen atoms emit radiation when they go back to the stable state. Eventually, the hydrogen would fall back to the lowest level, and perhaps, one day, it would react with the proton in nuclear decay.
I'm not clear on the other ideas you are talking about. I think what might be worth your time if you are serious about understanding this is investigating EM, QM, and eventually QFT. This would take years, of course, and require an understanding of advanced mathematics. But it would be worth it.