Accelerating charge generates electromagnetic waves and loses energy, in QFT terms it emits photons that carry it away. What of a static charge? Moving photons are usually associated with waves, which do not seem to be there in a static field, and static charge has no energy to lose. However, in the classical picture it fills the space with something, and another charge "feels" it when placed into it. So what exactly does it interact with locally?
I would have just made a comment, but I'm not allowed yet--
It sounds like your more general underlying question is: "How do we think about classical electromagnetic fields from the perspective of QED?" In particular, which configuration of quantum fields in QED do we associate with classical fields? How do we derive Maxwell's equations for these classical fields, using the QED equations of motion for the corresponding quantum fields? How do we derive the Lorentz force law describing the force on electrons?
The basic answer is that classical fields are best represented by coherent states in QED. These states are an infinite superposition of photon number states -- that is, there's nonzero probability of finding any number of photons. (In particular, probabilities for different photon numbers obey Poisson statistics.) The expected value of the electromagnetic field operators for these states corresponds to what we'd classically call the values of the electromagnetic field, and these expected values obey classical Maxwell equations.
The best notes I've found going through these questions might be: http://hitoshi.berkeley.edu/221b/photons.pdf.
You might also look at related SE questions What is the quantum state of a static electric field? and Given expectation values for E and B, can you find an associated state?.
A more direct answer to your immediate question: the classical field produced by a static charge corresponds to a quantum field (a coherent state) with high photon number. You point out that single photons are associated with traveling waves, yet somehow we have a static field. There's no contradiction because the coherent state is a superposition of many such traveling waves and thus has different behavior. Another charge placed nearby will interact with this photon field.