  The vector Apodizing Phase Plate (vAPP) is a class of pupil plane coronagraph
that enables high-contrast imaging by modifying the Point Spread Function (PSF)
to create a dark hole of deep flux suppression adjacent to the PSF core. Here,
we recover the known brown dwarf HR 2562 B using a vAPP coronagraph, in
conjunction with the Magellan Adaptive Optics (MagAO) system, at a
signal-to-noise of S/N = 3.04 in the lesser studied L-band regime. The data
contained a mix of field and pupil-stabilised observations, hence we explored
three different processing techniques to extract the companion, including
Flipped Differential Imaging (FDI), a newly devised Principal Component
Analysis (PCA)-based method for vAPP data. Despite the partial
field-stabilisation, the companion is recovered sufficiently to measure a 3.94
$\mu$m narrow-band contrast of (3.05$\pm$1.00) $\times$ 10$^{-4}$
($\Delta$m$_{3.94 {\mu}m}$ = 8.79$\pm$0.36 mag). Combined with archival GPI and
SPHERE observations, our atmospheric modelling indicates a spectral type at the
L/T transition with mass M = 29$\pm$15 M$_{\text{Jup}}$, consistent with
literature results. However, effective temperature and surface gravity vary
significantly depending on the wavebands considered
(1200$\leq$T$_{\text{eff}}$(K)$\leq$1700 and 4.0$\leq$log(g)(dex)$\leq$5.0),
reflecting the challenges of modelling objects at the L/T transition.
Observations between 2.4-3.2 $\mu$m will be more effective in distinguishing
cooler brown dwarfs due to the onset of absorption bands in this region. We
explain that instrumental scattered light and wind-driven halo can be
detrimental to FDI+PCA and thus must be sufficiently mitigated to use this
processing technique. We thus demonstrate the potential of vAPP coronagraphs in
the characterisation of high-contrast substellar companions, even in
sub-optimal conditions, and provide new, complementary photometry of HR 2562 B.
