  We demonstrate the formation of robust zero energy modes close to magnetic
impurities in the iron-based superconductor FeSe$_{1-x}$Te$_x$. We find that
the Zeeman field generated by the impurity favors a spin-triplet inter-orbital
pairing as opposed to the spin-singlet intra-orbital pairing prevalent in the
bulk. The preferred spin-triplet pairing preserves time-reversal symmetry and
is topological, as robust, topologically-protected zero modes emerge at the
boundary between regions with different pairing states. Moreover, the zero
modes form Kramers doublets that are insensitive to the direction of the spin
polarization or to the separation between impurities. We argue that our
theoretical results are consistent with recent experimental measurements on
FeSe$_{1-x}$Te$_x$.
