Can a charged particle emit light because of their intrinsic angular momentum even when they are stationary? Maxwell's laws state that an electric field in motion (here, possessing angular momentum) should produce a magnetic field and an electromagnetic wave (light). All charged particles with  spin (intrinsic angular momentum) can create a magnetic field even when they are stationary. They must also produce light because of this even when they are stationary.
 A: Charged particles emit photons only when accelerated assuming there is no other interaction with other particles (i.e. absorb  an external virtual photon from a nearby electron which would also mean that due to the conservation of momentum the charged particle would be accelerated). This means also that a constant velocity fermion will not emit any photons and being equivalent with the particle being at rest.
Why does accelerating electron emits photons?
http://www.tapir.caltech.edu/~teviet/Waves/empulse.html
A: One easy check you can consider is the fact that light is energy, and that it has to come from somewhere; if the particle isn’t accelerating and has no decaying or anything, it would have to come from the spin, and so the spin would have to slow down so whatever mechanism would allow light to be emitted. That wouldn’t be an indefinitely sustainable process, as eventually you run out of energy and particles would stop spinning, and we know things don’t just stop spinning, so therefore that can’t be a process with those specific limitations
A: Just to add a little to existing answers.
It is helpful to divide physics up into 'classical physics' and 'quantum physics'. Quantum physics is a model which includes classical physics as a kind of subset.
According to classical physics, you can have a current which produces a magnetic field without producing any emitted radiation. An ordinary current-carrying wire is an example, if the current is constant. If the wire makes a loop then you have a loop of current and the magnetic field far from the loop has the same form as the one produced by a charged particle with spin.
Quantum physics says the same (as it always does whenever any statement from classical physics proves to be correct!) but quantum physics allows us to grasp what is going on for the charged particle with spin in more detail. It is subtle because the presence of the property called 'spin' is not the same as having rotation in the ordinary sense. Spin is a special form of angular momentum which does not require any matter to be in motion. When a particle has both charge and spin then it has magnetic properties, and can produce a magnetic field. The situation is then a bit like my example of the loop of current-carrying wire, but not quite like. The main thing is that the ability to be the source of a field is not the same as producing electromagnetic radiation. But if something which produced a field is also made to accelerate, then typically you do get emitted radiating energy. A particle sitting still is not accelerating, even though it may have the property called spin (or 'intrinsic angular momentum' to use the more technical name.)
A: 
According to classical physics, you can have a current which produces a magnetic field without producing any emitted radiation.

The electrons on the wire do not move forward continuously, but constantly bump into atomic corpses and other electrons. This manifests itself through the electrical resistance of the conductor. The power loss is clearly emitted as radiation. In addition to the creation of a common magnetic field, EM radiation is generated - although largely unnoticed by the experimenter. For this reason, it is interesting to observe what happens when electrons flow into a vacuum tube. Does a common magnetic field also develop here? Unfortunately I do not know of any such experiments

Quantum physics … allows us to grasp what is going on for the charged particle with spin in more detail. It is subtle because the presence of the property called 'spin' is not the same as having rotation in the ordinary sense. Spin is a special form of angular momentum which does not require any matter to be in motion.

Spin entered physics when it became clear that the characterisation of the electron in the atom was incomplete. It was a question of distinguishing electrons, which in pairs have almost the same state in the atom. This property was called spin and clearly related to the behaviour of the electron in a magnetic field, in which it is known to be deflected when moving through it.
At the same time, it was discovered that the electron not only has an electric charge, but is also a magnet. No one had the idea of conceding both fields to the electron as immanent, and so the term spin has been used ever since, when it is actually more intuitive to speak of the magnetic dipole of the electron.

When a particle has both charge and spin then it has magnetic properties, and can produce a magnetic field.

This is more complicated than if I simply grant the electron an intrinsic (present in all circumstances) magnetic dipole. The sum of the commonly aligned electrons manifests itself in the common (macroscopic) magnetic field.

But if something which produced a field is also made to accelerate, then typically you do get emitted radiating energy.

I see exactly two cases where electrons emit EM radiation. The first case is trivial and has to do with the fact that it is impossible - our experience has made this a law - to transmit energy without loss. So if I move electrons in a wire, part of the energy put into it is emitted as power loss in the form of EM radiation.
The second case is interesting. When an electron passes through a magnetic field, it is deflected perpendicular to the plane of the magnetic field and the direction of movement. It runs through a spiral path and loses kinetic energy in the process, which is noticeable as EM radiation. And now it gets really interesting.
What if the radiation is the reason for the deflection of the electron into a spiral path? The Lorentz force is a law that was determined empirically and does not require an explanation of the process of deflection. It does not even take into account the loss of energy that inevitably occurs.
The process of deflection in the magnetic field can be explained if one considers the alignment of the magnetic dipole of the electron in the external magnetic field and mentally allows that a photon is ejected in the process that both disturbs this alignment again and changes the direction of the electron. The photon carries energy and momentum away, and the electron changes its direction (momentum) and is slowed down in its movement (kinetic energy).

A particle sitting still is not accelerating, even though it may have the property called spin (or 'intrinsic angular momentum' to use the more technical name.)

NIST calls it the electron magnetic moment and it is considered a fundamental physical constant.
Can a charged particle emit light because of their intrinsic angular momentum even when they are stationary? Can not. Spin is a synonym for the intrinsic magnetic moment of the electron and really has nothing to do with a rotation.
Thanks to Andrew Steane for the basis to answer the question.
