Dark matter particles, even if they are electrically neutral, could interact with the Standard Model particles via their electromagnetic multipole moments. In this paper, we focus on the electromagnetic properties of the complex vector dark matter candidate, which can be described by means of seven form factors. We calculate the differential scattering cross-section with nuclei due to the interactions of the dark matter and nuclear multipole moments, and we derive upper limits on the former from the non-observation of dark matter signals in direct detection experiments. We also present a model where the dark matter particle is a gauge boson of a dark SU(2) symmetry, and which contains heavy new fermions, charged both under the dark SU(2) symmetry and under the electromagnetic U(1) symmetry. The new fermions induce at the one loop level electromagnetic multipole moments, which could lead to detectable signals in direct detection experiments.