  Scanning tunneling experiments on single crystals of superconducting
FeTe$_{0.55}$Se$_{0.45}$ have recently provided evidence for discrete energy
levels inside vortices. Although predicted long ago, such levels are seldom
resolved due to extrinsic (temperature, instrumentation) and intrinsic
(quasiparticle scattering) limitations. We study a microscopic multiband model
with parameters appropriate for FeTe$_{0.55}$Se$_{0.45}$. We confirm the
existence of well-separated bound states and show that the chemical disorder
due to random occupation of the chalcogen site does not affect significantly
the vortex-core electronic structure. We further analyze the vortex bound
states by projecting the local density of states on angular-momentum
eigenstates. A rather complex pattern of bound states emerges from the
multiband and mixed electron-hole nature of the normal-state carriers. The
character of the vortex states changes from hole-like with negative angular
momentum at low energy to electron-like with positive angular momentum at
higher energy within the superconducting gap. We show that disorder in the
arrangement of vortices most likely explains the differences found
experimentally when comparing different vortices.
