Action at a distance in Quantum Field Theory

No. And the example you provide is not action at a distance. As you stated, there is an exchange of virtual photons. Quantum field theory has laws that come from symmetries (like all of physics). Sometimes these symmetries are global or local.

In modern QFT it is believed that global symmetries are unnatural. They have an “action at a distance” feel. We now suspect that all fundamental symmetries are local gauge symmetries. Global symmetries are either all broken (such as parity, time reversal invariance, and charge symmetry) or approximate (such as isotopic spin invariance) or they are the remains of spontaneously broken local symmetries.

In the case of a certain physical phenomena that arises from a global symmetry, it is necessary that this same physical symmetry has to hold locally. That is, the symmetry must be preserved locally. In fact it is demanded that certain physical phenomena and the mathematics defining these physical phenomena therein have to be refined so that locally the same physics is preserved. This is necessary so as to discount “action at a distance” which is not physical. For example in the Higgs mechanism a mass term arises which breaks gauge symmetry. For this we add a scalar field term that couples to the gauge field via the addition of a covariant derivative.