In copper- and iron-based unconventional superconductors, the Bogoliubov quasiparticles interact with a spin resonance at momentum $(pi,pi)$. This interaction is revealed by specific signatures in the quasiparticle spectroscopies, like kinks in photoemission and dips in tunneling. We study these signatures, as they appear inside and around a vortex core in the local density of states (LDOS), a property accessible experimentally by scanning tunneling spectroscopy. Our model retains the whole nonlocal structure of the self-energy in space and time and is therefore not amenable to a Hamiltonian treatment using Bogoliubov--de Gennes equations. The interaction with the spin resonance does not suppress the zero-bias peak at the vortex center, although it reduces its spectral weight; neither does it smear out the vortex LDOS, but rather it adds structure to it. Some of the signatures we find may have been already measured in FeSe, but remained unnoticed. We compare the LDOS as a function of both energy and position with and without coupling to the spin resonance and observe, in particular, that the quasiparticle interference patterns around the vortex are strongly damped by the coupling. We study in detail the transfer of spectral weight induced both locally and globally by the interaction and also by the formation of the vortex. Finally, we introduce a new way of imaging the quasiparticles in real space, which combines locality and momentum-space sensitivity. This approach allows one to access quasiparticle properties that are not contained in the LDOS.