  VHS J1256$-$1257 AB is an ultracool dwarf binary that hosts a wide-separation
planetary-mass companion that is a key target of the {\sl JWST} Exoplanet Early
Release Science (ERS) program. Using Keck adaptive optics imaging and aperture
masking interferometry, we have determined the host binary's orbit,
$a=1.96\pm0.03$ au, $P=7.31\pm0.02$ yr, $e=0.883\pm0.003$, and measured its
dynamical total mass, $0.141\pm0.008$ $M_{\odot}$. This total mass is
consistent with VHS J1256$-$1257 AB being a brown dwarf binary or pair of very
low-mass stars. In addition, we measured the orbital motion of VHS J1256$-$1257
b with respect to the barycenter of VHS J1256$-$1257 AB, finding that the wide
companion's orbit is also eccentric, $e=0.68^{+0.11}_{-0.10}$, with a mutual
inclination of $115\pm14^{\circ}$ with respect to the central binary. This
orbital architecture is consistent with VHS J1256$-$1257 b attaining a
significant mutual inclination through dynamical scattering and thereafter
driving Kozai-Lidov cycles to pump the eccentricity of VHS J1256$-$1257 AB. We
derive a cooling age of $140\pm20$ Myr for VHS J1256$-$1257 AB from low-mass
stellar/substellar evolutionary models. At this age, the luminosity of VHS
J1256$-$1257 b is consistent with both deuterium-inert and deuterium-fusing
evolutionary tracks. We thus find a bimodal probability distribution for the
mass of VHS J1256$-$1257 b, either $12.0\pm0.1$ $M_{\rm Jup}$ or $16\pm1$
$M_{\rm Jup}$, from these models. Future spectroscopic data to measure
isotopologues such as HDO and CH$_3$D could break this degeneracy and provide a
strong test of substellar models at the deuterium-fusion mass boundary.
