What is the density of virtual photons around a unit charge? It seems that virtual photons also exist in vacuum. So the precise question is:
What is the additional virtual photon density due to the electric field of a unit charge?
Or: How many virtual photons per volume are found around a unit charge?
The answer will depend on distance, but what are the exact numbers?
 A: I do not think that this question has an answer. A photon is a quantum mechanical object.
a) there is no conservation of photons, virtual or not. 
b)there is no lower limit to the energy of the photons, so in principle they are infinite ( infrared problem)
c)The energy of the virtual photons will depend on the motion of the charge and or the probe that will be checking for virtual photons.
You can have an energy spectrum for virtual photons, but not a density estimate . See this caluclation for example.
A: It's infinity. This is the soft photon problem, which requires infrared regularization.
A: Apparently you imagine a charge as a point with a Coulomb field around it. Outside the charge there is no charge but there is a field, you say, and it consists of virtual photons. So how many of them are at the distance $r$ from the charge?
I let the others answer this question and here I will give you my vision of that. The charge is not point-like but quantum mechanically smeared. It does not stay at a point but "moves" and creates a quantum mechanical "cloud". This "cloud" is "large and soft" (like a jelly) - you push the charge and the whole system gets deformed inelastically. You break the original state and the new state is a moving now charge and excited quantum oscillators (oscillating cloud). The latter describes photons.  So normally the initial state is different from the final state. In experiment they usually add together all particular "pictures" of all scattering events and obtain an "inclusive" picture. It is the inclusive picture that corresponds to the point-like image of the charge.
