  Extreme mass ratio inspirals (EMRIs) are excellent sources for space-based
observatories to explore the properties of black holes and test no-hair
theorems. We consider EMRIs with a charged compact object inspiralling onto a
Kerr black hole in quasi-circular orbits. Using the Teukolsky and generalized
Sasaki-Nakamura formalisms for the gravitational and vector perturbations about
a Kerr black hole, we numerically calculate the energy fluxes for both
gravitational and vector perturbations induced by a charged particle moving in
equatorial circular orbits. With one-year observations of EMRIs, we apply the
Fisher information matrix method to estimate the charge uncertainty detected by
space-based gravitational wave detectors such as the Laser Interferometer Space
Antenna, TianQin, and Taiji, and we find that it is possible to detect vector
charge as small as $q\sim 0.0049$. The results show that EMRIs composed of a
Kerr black hole with a higher spin $a$ and lighter mass $M$, and a secondary
charged object with more vector charge give smaller relative error on the
charge, thus constrain the charge better. The positive spin of the Kerr black
hole can decrease the charge uncertainty by about one or two orders of
magnitude.
