why don't other field forces like magnetic fields have their elementary particles?
The source of electric fields are electric charges. So the subatomic particles electron and proton are sources of electric fields. To observe an electric field negative and positive charge(s) have to be separated.
Somehow the same one can say about magnetic fields. The subatomic particles obey a magnetic dipole moment. This is an intrinsic (existing independent from some circumstances) property of these particles. To observe an magnetic field the involved particles have to be aligned with their magnetic dipole moments.
If one considers the magnetic field to be a special type of EM field with 0 amplitude electric field, then should you expect to detect a photon when you place a photon detector near a magnetic field?
In the case an electron approaches to a nucleus we indeed can observe the emission of photons. Another example for the emission of photons is the acceleration of electrons, best observable in an antenna rod where electrons get accelerated back and forth the rod. The interesting fact is that if one observe this radiation from an antenna the radiation is composed of an electric and a magnetic field. So an electron - with its electric charge and its magnetic dipole moment - emits EM radiation. But you would not be able to detect a photon near an electric nor near a magnetic field.
To describe the interaction of electric fields among themselves, magnetic fields among themselves and moving charges in magnetic fields one uses the construct of virtual photons. How this interaction happens in a closer view isn't in the focus of today's physic.