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 a vortex can serve as a platform for phase-sensitive measurements to answer this question. By solving the 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. At energies closer to the gap value, however, the LDOS can distinguish an anisotropic s-wave gap 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 a vortex can therefore be used as a real-space probe of the gap structure. © IOP Publishing and Deutsche Physikalische Gesellschaft.