  A monolayer of the high-$T_c$ superconductor FeTe$_{1-x}$Se$_x$ has been
predicted to realize a topologically non-trivial state with helical edge modes
at its boundary, providing a novel intrinsic system to search for topological
superconductivity and Majorana zero modes. Evidence in favor of a topological
phase transition and helical edge modes has been identified in recent
experiments \cite{Peng2019}. We propose to create Majorana zero modes by
applying an in-plane magnetic field to the FeTe$_{1-x}$Se$_x$ monolayer and by
tuning the local chemical potential via electric gating. Owing to the
anisotropic magnetic couplings on edges from a topological band inversion, an
in-plane magnetic field drives the system into an intrinsic high-order
topological superconductor phase with Majorana corner modes, without
fabricating heterostructures. Furthermore, we demonstrate that Majorana zero
modes can occur at other different locations, including the domain wall of
chemical potentials at one edge and certain type of tri-junction in the 2D
bulk. Our study not only demonstrates FeTe$_{1-x}$Se$_x$ monolayer as a
promising Majorana platform with scalability and electrical tunability and
within reach of contemporary experimental capability, but also provides a
general principle to search for realistic realization of high-order topological
superconductivity.
