  We have performed device-based tunnelling spectroscopy of NbSe$_2$ in the
vortex state with a magnetic field applied both parallel and perpendicular to
the $a-b$ plane. Our devices consist of layered semiconductors placed on top of
exfoliated NbSe$_2$ using the van der Waals transfer technique. At zero field,
the spectrum exhibits a hard gap, and the quasiparticle peak is split into low
and high energy features. The two features, associated with the effective
two-band nature of superconductivity in NbSe$_2$, exhibit markedly distinct
responses to the application of magnetic field, suggesting an
order-of-magnitude difference in the spatial extent of the vortex cores of the
two bands. At energies below the superconducting gap, the hard gap gives way to
vortex-bound Caroli-de Gennes-Matricon states, allowing the detection of
individual vortices as they enter and exit the junction. Analysis of the
sub-gap spectra upon application of parallel magnetic field allows us to track
the process of vortex surface formation and spatial rearrangement in the bulk.
