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 theMagellan 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-stabilized 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 fieldstabilization, the companion is recovered sufficiently to measure a 3.94 μm narrow-band contrast of (3.05 ± 1.00) × 10-4 (Δm3.94μm = 8.79±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±15MJup, consistent with literature results. However, effective temperature and surface gravity vary significantly depending on the wavebands considered (1200 ≤ Teff(K) ≤ 1700 and 4.0 ≤ log(g)(dex) ≤ 5.0), reflecting the challenges of modelling objects at the L/T transition. Observations between 2.4 and 3.2 μm 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 characterization of high-contrast substellar companions, even in sub-optimal conditions, and provide new complementary photometry of HR 2562 B.