  The structure of a vortex in the inner crust of a pulsar is calculated
microscopically in the Wigner-Seitz cell approximation, simulating the
conditions of the inner crust of a cold, non-accreting neutron star, in which a
lattice of nuclei coexists with a sea of superfluid neutrons. The calculation
is based on the axially deformed Hartree-Fock-Bogolyubov framework, using
effective interactions. The present work extends and improves previous studies
in four ways: i) it allows for the axial deformation of protons induced by the
large deformation of neutrons due to the appearance of vortices; ii) it
includes the effect of Coulomb exchange; iii) considers the possible effects of
the screening of the pairing interaction; and iv) it improves the numerical
treatment. We also demonstrate that the binding energy of the nucleus-vortex
system can be used as a proxy to the pinning energy of a vortex and discuss in
which conditions this applies. From our results, we can estimate the mesoscopic
pinning forces per unit length acting on vortices. We obtain values ranging
between $10^{14}$ to $10^{16}$ dyn/cm, consistent with previous findings.
