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.
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