We consider electromagnetically neutral dark states that couple to the photon through higher dimensional effective operators, such as electric and magnetic dipole moment, anapole moment and charge radius operators. We investigate the possibility of probing the existence of such dark states, taking a Dirac fermion $\chi$ as an example, at several representative proton-beam experiments. As no positive signal has been reported, we obtain upper limits (or projected sensitivities) on the corresponding electromagnetic form factors for dark states lighter than several GeV. We demonstrate that while the current limits from proton-beam experiments are at most comparable with those from high-energy electron colliders, future experiments, such as DUNE and SHiP, will be able to improve the sensitivities to electric and magnetic dipole moment interactions, owing to their high intensity.