Majorana quasiparticles in condensed matter are important for topological quantum computing(1-3), but remain elusive. Vortex cores of topological superconductors may accommodate Majorana quasiparticles that appear as the Majorana bound state (MBS) at zero energy(4,5). The iron-based superconductor Fe(Se,Te) possesses a superconducting topological surface state(6-9) that was investigated by scanning tunnelling microscopy (STM) studies, which suggest such a zero-energy vortex bound state (ZVBS)(10,11). Here we present ultrahigh energy-resolution spectroscopic imaging (SI)-STM to clarify the nature of the vortex bound states in Fe(Se,Te). We found the ZVBS at 0 +/- 20 mu eV, which constrained its MBS origin, and showed that some vortices host the ZVBS but others do not. We show that the fraction of vortices hosting the ZVBS decreases with increasing magnetic field and that local quenched disorders are not related to the ZVBS. Our observations elucidate the necessary conditions to realize the ZVBS, which paves the way towards controllable Majorana quasiparticles.