  The kinetic Sunyaev Zel'dovich effect is a secondary CMB temperature
anisotropy that provides a powerful probe of the radial-velocity field of
matter distributed across the Universe. This velocity field is reconstructed by
combining high-resolution CMB measurements with galaxy survey data, and it
provides an unbiased tracer of matter perturbations in the linear regime. In
this paper, we show how this measurement can be used to probe primordial
non-Gaussianity of the local type, particularly focusing on the trispectrum
amplitude $\tau_{\rm NL}$, as may arise in a simple two-field inflation model
that we provide by way of illustration. Cross-correlating the
velocity-field-derived matter distribution with the biased large-scale galaxy
density field allows one to measure the scale-dependent bias factor with sample
variance cancellation. We forecast that a configuration corresponding to CMB-S4
and VRO results in a sensitivity of $\sigma_{f_{\rm NL}} \approx 0.59$ and
$\sigma_{\tau_{\rm NL}} \approx 1.5$. These forecasts predict improvement
factors of 10 and 195 for $\sigma_{f_{\rm NL}}$ and $\sigma_{\tau_{\rm NL}}$,
respectively, over the sensitivity using VRO data alone, without internal
sample variance cancellation. Similarly, for a configuration corresponding to
DESI and SO, we forecast a sensitivity of $\sigma_{f_{\rm NL}} \approx 3.1$ and
$\sigma_{\tau_{\rm NL}} \approx 69$, with improvement factors of 2 and 5,
respectively, over the use of the DESI data-set in isolation. We find that a
high galaxy number density and large survey volume considerably improve our
ability to probe the amplitude of the primordial trispectrum for the
multi-field model considered.
