A major question in Fe-based superconductors remains the structure of the
pairing, in particular whether it is of unconventional nature. The electronic
structure near vortices can serve as a platform for phase-sensitive
measurements to answer this question. By solving Bogoliubov-de Gennes equations
for LiFeAs, we calculate the energy-dependent local electronic structure near a
vortex for different nodeless gap-structure possibilities. At low energies, the
local density of states (LDOS) around a vortex is determined by the
normal-state electronic structure. However, at energies closer to the gap
value, the LDOS can distinguish an anisotropic from a conventional isotropic
s-wave gap. We show within our self-consistent calculation that in addition,
the local gap profile differs between a conventional and an unconventional
pairing. We explain this through admixing of a secondary order parameter within
Ginzburg-Landau theory. In-field scanning tunneling spectroscopy near vortices
can therefore be used as a real-space probe of the gap structure