Electromagnetically neutral dark sector particles may feebly interact with photons through higher dimensional effective operators, such as mass-dimension 5 magnetic and electric dipole moment, and a mass-dimension 6 anapole moment and charge radius operators. In this work, we use hypercharge gauge field form factors to treat dark states, which will induce not only electromagnetic form factors but also the corresponding $Z$ boson operators. Taking a Dirac fermion $χ$ as an example, we investigate the probes of searching for such dark states at current and future $e^+e^-$ collider experiments including BESIII, STCF, Belle II and CEPC via monophoton searches. Comparing to current experiments, we find that electron colliders including BESIII, STCF, Belle II, which operate with the center-of-mass at several GeV, have leading sensitivity on the corresponding electromagnetic form factors for the mass-dimension 5 operators with dark states lighter than several GeV, while they can not provide competitive upper limits for the mass-dimension 6 operators. Future CEPC operated with the center-of-mass on and beyond the mass of $Z$-boson with competitive luminosity can probe the unexplored parameter space for dark states with mass-dimension 5 (6) operators in the mass region of $m\lesssim 100 $ GeV (10 MeV $\lesssim m\lesssim 100 $ GeV).

17 pages, 5 figures. Matched the published version in Phys. Rev. D