  We construct a scotogenic Majorana neutrino mass model in a gauged $U(1)_X$
extension of the standard model, where the mass of the gauge boson and the
unbroken gauge symmetry, which leads to a stable dark matter (DM), can be
achieved through the Stueckelberg mechanism. It is found that the simplest
version of the extended model consists of the two inert-Higgs doublets and one
vector-like singlet fermion. In addition to the Majorana neutrino mass, we
study the lepton flavor violation (LFV) processes, such as $\ell_i \to \ell_j
\gamma$, $\ell_i \to 3 \ell_j$, $\mu-e$ conversion rate in nucleus, and
muonium-antimuonium oscillation. We show that the sensitivities of $\mu\to 3e$
and $\mu-e$ conversion rate designed in Mu3e and COMET/Mu2e experiments make
both decays the most severe constraints on the $\mu\to e$ LFV processes. It is
found that $\tau\to \mu \gamma$ and $\tau\to 3\mu$ can reach the designed
significance level of Belle II. In addition to explaining the DM relic density,
we also show that the DM-nucleon scattering cross section can satisfy the
currently experimental limit of DM direct detection.
