What is the source of the electric charge on the electron? An individual electron possesses an electric charge.  Can the existence of this charge be explained further, or is this simply a current "fundamental" of physics?
 A: This is the table of elementary particles in the standard model of particle physics

So it is not just the electrons that are posited as  axiomatic assumption of the model, but all the particles in the list. 
These with the SU(3)xSU(2)xU(1) symmetries and the solutions of the corresponding differential equations assure that the plethora of data in particle physics fit, and predictions of future data work, as seen at LHC experiments.
So it is an experimental assumption. Note that it is mass, charge and spin that are assumed, together with the other quantum numbers. Not magnetic moments, these arise by the solution of the differential equations describing these particles.
In general, classical electromagnetic fields can be shown to emerge smoothly from the quantum mechanical level that underlies all of nature, but it needs quantum electrodynamics to understand the process.
A: In classical physics, we often say the charge of an electron is what makes it interact with the electromagnetic field. In quantum field theory, the charge is sort of "explained", in the sense that these two phenomena turn out to be the exact same thing! The electron field interacts with the electromagnetic field (where "interacts" just means "costs some energy to be in the same place") and from this interaction we can derive that the excitations of the electron field have charge, as well as a magnetic moment, and so on. That's the most fundamental explanation we know of electromagnetic phenomena.
You're right that we can't derive the strength of the coupling between the fields -- we have to measure it, often through a simple proxy like the charge of the electron. But once we have that, everything else is fixed.
A: A
The article "Is the electron a photon with toroidal topology?" dated 1996 or 1993 proposes the charge emerge from the photon which become electron by making two loops within one wavelength. Looping cause the photon lost it helicity and electric field become radially and magnetic field axially oriented. There is no peer reviewed paper referencing this work. On the other hand some recent experiments shows a photon can do curves using near field (evanescent waves).
A: 
What is the source of the electric charge on the electron?

Geometry. Google on Percy Hammond electromagnetism.

An individual electron possesses an electric charge. 

Note though that an electron's field is an electromagnetic field. See section 11.10 of Jackson's Classical Electrodynamics where he says "one should properly speak of the electromagnetic field $F_{\mu\nu}$ rather than E or B separately". You can't separate the electron's electromagnetic field into an electric field and a magnetic field, so IMHO it's better to think in terms of electromagnetic charge 

Can the existence of this charge be explained further, or is this simply a current "fundamental" of physics?

It can be explained further. See Sudanamaru's reference to Is the electron a photon with toroidal topology? and note the mention of two loops. Also note Dirac's belt on mathpages: "In this sense a Mobius strip is reminiscent of spin-½ particles in quantum mechanics, since such particles must be rotated through two complete rotations in order to be restored to their original state". Then note that field is the derivative of potential, and depict the four-potential as the integral of the electromagnetic sine wave: 

Then wrap up your field-variation into a chiral "Mobius" spin ½ spinor, like this:

The major and minor components of the field variation now combine into an all-round standing field. Remember it's the wave nature of matter, and remember this: standing wave, standing field.      
