In recent years, many clever realizations of Majorana fermions in condensed matter have been predicted -- and some largely verified -- by exploiting the interplay between superconductivity and band topology in metals and insulators. However, realizations in semimetals remain less explored. We ask, under what conditions do superconductor vortices in time-reversal symmetric Weyl semimetals trap Majorana fermions on the surface? If each constant-$k_{z}$ plane, where $z$ is the vortex axis, contains equal numbers of Weyl nodes of each chirality, we predict a generically gapped vortex and derive a topological invariant $ u$ in terms of the Fermi arc structure that signals the presence or absence of surface Majorana fermions. In contrast, if certain constant-$k_{z}$ planes contain a net chirality of Weyl nodes, the vortex is gapless. We analytically calculate $ u$ within a perturbative scheme and provide numerical support with an orthorhombic lattice model. Using our criteria, we predict phase transitions between trivial, critical and topological vortices by simply tilting the vortex, and propose Li(Fe$_{0.91}$Co$_{0.09}$)As with broken inversion symmetry as a candidate for realizing our proposals.
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