How to calculate number of exchanged virtual photons per unit of time between two electromagnetically interacting objects?
Imagine an elastic collision of two protons - something that they also see at the LHC but it's not the most interesting kind of interaction.
The two protons will repel because of the electromagnetic interaction. Assume that the distance between them is never too small, relatively to the proton radius. But you may calculate the cross section of this process in quantum field theory in different ways.
You may treat protons as elementary particles - which is OK if you don't break them to pieces and if you don't study their composition in any way. If that's so, it's directly a proton that emits the virtual photon, and it is absorbed by the other photon.
So at the leading order - which is pretty accurate - the process only involves a single virtual proton - per the whole interaction. If you interpret the proton as a collection of three quarks, of course there will be many more interactions and each quark will interact with the quarks in the other proton - as well as its friends in the same proton - so there will be many more virtual particles.
But the very fact that I could obtain the result "one" for the number of virtual protons that are exchanged should convince you that you may be imagining the "exchange of the virtual photons" too naively. A virtual photon is simply not a particle that objectively exists and that you can count. A virtual photon is a part of a mathematical expression - the Feynman diagram - that is used to evaluate the only thing that makes a physical sense here, namely the probability amplitude (producing the cross section) for a particular process.
The number of virtual particles - the propagators - depends on the Feynman diagram. And you must actually sum all the Feynman diagrams (it's coming from the sum over different histories) - and each of them gives a different answer to the question how many virtual particles there have been - before you get the probability amplitude.
Moreover, even in "individual diagrams", the number of virtual particles depends on the resolution you choose - whether you try to decompose particles into their internal structure or not.
In fact, even if you collide two charged cellestial bodies, you may still say that they exchanged 1 virtual photon only. The virtual photons are just not real objects that you may count as apples. Only the final measurements are real - anything that happened in between is virtual and all histories are allowed and all histories contribute - and even the final measurements are probabilistic in character.
So even if you were trying to count the number of physical, real particles in a state, you could get misleading, divergent, and ambiguous numbers. The counting of the virtual particles is even much more ill-defined.
The short answer is that you just can't. Different ways of doing the accounting (e.g., different gauges) lead to different ways of counting up the virtual particles in any given situation.