  Honeycomb cobalt oxides containing 3$\it{d}$ Co$^{2+}$ ions might realize
frustrated magnetism and novel quantum phases. Among candidate materials,
Na$_{3}$Co$_{2}$SbO$_{6}$ stands out for its distorted honeycomb lattice and
significant in-plane anisotropy, motivating vector-field tuning inside the
honeycomb plane. Here we use thermal transport down to the mK regime to study
twin-free crystals of Na$_{3}$Co$_{2}$SbO$_{6}$ subject to in-plane vector
fields. We find that the thermal conductivity $\kappa$ never exceeds the
heat-transport capability of phonons, rendering its suppression primarily due
to phonon scattering off magnetic excitations and/or domain boundaries. The
system's field-driven quantum criticality manifests itself as an abundance of
magnetic fluctuations hindering the heat transport, which further depends on
the field direction in an intriguing manner.
