Quantum fields interact in a quantum way ("digital" instaed of "analog"). Two charged particles both have quantum fields. These fields are NOT the electromagnetic field. Quantum fields are not real in the sense that they cannot be directly measured or observed. They represent the probability of the interaction. The fields don't constantly "feel" each other, do not constantly attract, repulse or interact in any way. Then at some random point (with the probablity defined by the fields) a momentary interaction happens, after which the fields again do not "feel" each other. This interaction tranfers energy, momentum, and certain quantum numbers (depending on the type of interaction this could be charge, "color", etc.). The values transferred by the interaction (except for the mass) look like they could belong to a particle, but this particle does not actually exist. Therefore, for simplicity, sometimes the interaction is viewed as an exchange of a virtual particle, where "virtual" as opposed to "real" emphasizes that this particle does not exist, but is just a way to combine the transferred values into a single hypothetical object.
Referring to your question, this interaction (or a "virtual particle", if you will) is an electromagnetic interaction and a part of the electromagnetic field. When you have many of such interactions, their quantum ("digital") nature averages out and you see a continuous ("analog") classical electromagnetic field. In the sense described above you can hypothetically say that the electromagnetic field consists of "virtual photons", as long as you remember that they don't actually exist, but are only a convenient way to decsribe how the quantum fields interact.
As you hopefully can see, virtual photons are not emitted constantly, but only "on demand" when the quantum fields of two particles "decide" to interact at a random point, but on average based on the probabilities defined by the intencity of the fields. Once again to emphasize, the quantum fields do not directly represent the electromagnetic field. The electromagnetic field is represented by a statistical sum of a large number of interactions between the quantum fields. And in a simplified way these interactions sometimes can be viewed as exchanges of "virtual particles", as long as we remember that they are not real and therefore do not actually exist. So in this view, one could say that the electromagnetic field consists of virtual photons as hypothetical objects representing quantum interactions between the quantum fields.
In contrast with the electromagnetic field, the electromagnetic wave can be viewed as consisting of real photons, but this is a different topic.