We propose an experimental search for an axion-induced oscillating electric dipole moment (OEDM) for electrons using state-of-the-art alkali vapor-cell atomic magnetometers. The axion is a hypothesized new fundamental particle which can resolve the strong charge-parity problem and be a prominent dark matter candidate. This experiment utilizes an atomic magnetometer as both a source of optically polarized electron spins and a magnetic-field sensor. The interaction of the axion field, oscillating at a frequency equal to the axion mass, with an electron spin induces a sizable OEDM of the electron at the same frequency as the axion field. When the alkali vapor is subjected to an electric field and a magnetic field, the electron OEDM interacts with the electric field, resulting in an electron spin precession at the spins Larmor frequency in the magnetic field. The resulting precession signal can be sensitively detected with a probe laser beam of the atomic magnetometer. We estimate that the experiment is sensitive to the axion-photon interaction in ultralight axion masses from $10^{-15}$ to $10^{-10}$~eV. It is able to improve the current experimental limit up to 5 orders of magnitude, exploring new axion parameter spaces.
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