Hyperbolic phonon polaritons (HPhPs) in low-loss anisotropic materials such as hexagonal boron nitride (hBN) provide extreme sub-diffractional field confinement in the mid-infrared, making them attractive for enhancing the interaction of light with molecular vibrations. Collective vibrational strong coupling between molecular ensembles and HPhPs has already been demonstrated in extended slabs. Reaching this regime with a single molecule, however, remains a major challenge, and would provide a platform for investigating how vacuum electromagnetic fields influence molecular dynamics and chemical reactivity. Here, we show theoretically that single-molecule vibrational strong coupling can be reached in hBN nanocavities that laterally confine HPhPs within a finite volume. Combining full-wave electromagnetic simulations with the few-mode quantization approach, we find that these nanocavities support a rich set of asymmetric resonances that strongly enhance the local spectral density, with frequencies that follow from simple HPhP ray trajectories and can be tuned through the cavity aspect ratio. We identify resonances with quality factors of about 600 and effective mode volumes of |V_eff| ∼ 10⁻⁷λ³, sufficient for single-molecule strong coupling with vibrational transitions with dipole moments on the order of 1 D. These results establish hyperbolic nanocavities as a promising platform for single-molecule vibrational strong coupling and open new routes toward the local control of vibrational dynamics and molecular energy flow.