First, virtual photons are not real, they are a mathematical method of describing the (in your case) stationary electric field around the electron.
This field exist around the electron. It is something that QM/QFT tries to describe mathematically, because in real particles terms, they are not describable.
Virtual photons do not obey some laws, for example they do not obey the speed limit c.
Virtual photons are a way of describing the interaction between the electron (charge) field and the other particles' fields, for example a proton's field (strong force field because of quarks make up the proton). The proton field interacts with the electron field by exchanging virtual photons. This is how protons and electrons attract.
But this interaction, and the exchange of virtual photons is what confuses you. This is not a real particle exchange. There are no real particles in this case exchanged.
Now this interaction is described in math as the exchange of virtual photons. But we do not know whether something is actually exchanged between a proton and the electron.
We do know that the proton and the electron do interact electromagnetically (and gravitationally too).
Now this EM interaction between the proton and the electron is what causes them to attract. We describe this interaction with the exchange of virtual photons. These are not real photons. They are off mass shell. They do not obey the speed limit c.
Now you are asking how many virtual photons are exchanged between the proton and the electron. You are confused because you learned that EM energy is quantized. The smallest piece of this EM energy is the real photon. Not the virtual photon.
Both the proton and the electron have a static EM field around them. Your question is whether this static EM field is quantized.
Now even if you would try to measure this static field around the charge, and then divide it into pieces, and call them virtual photons, this would not be real. In reality, the electron or the proton, neither of them are emitting anything in this case.
The static EM field around the charges is not created by emitting real particles. We cannot count them, because no real particle is emitted.
What we do know, is that the static EM field around the charge is somewhat similar to gravity. Let's see how similar.
Now stress-energy is the source of gravity. This creates a gravitational field around a mass. How many virtual gravitons does the Sun emit instantly, per time? Similarly, the Sun does not emit virtual gravitons. This is just a theoretical (and experimentally not yet proven) way to describe the gravitational effects of stress-energy (mass).
So how do we explain that gravity (stress-energy) interacts with other particle's fields? We say that gravity bends spacetime. This is the way we describe it instead of virtual gravitons. But the reality is, that gravity has an effect on the fabric of spacetime, so that anything that interacts with the gravitational field, will be affected by it.
Similarly, when there is a charge, like an electron in your case, and it has a static EM field around it, anything that interacts electromagnetically, will be affected by it. Do we say similarly that EM bends spacetime? Actually we do. But it is indirect. What we can say is that the charge has an effect on the fabric of spacetime and anything that interacts with the EM field will be affected by it.
So just as stress-energy (or the mass of the Sun) in reality does not emit virtual gravitons, and we cannot count the virtual gravitons, similarly, we say that in reality the electron in your case does not emit virtual photons, and we cannot count them.
So both the electron and the proton create a field that has an effect on the fabric of spacetime, and these fields interact with each other. This is how they attract. Mathematically we describe them as an exchange of virtual photons. In reality, no real particle is exchanged.
The EM field is quantized, but virtual photons are not the real quantization of the static EM field around the electron and the proton.