  Local non-Gaussianities in the initial conditions of the Universe,
parameterized by $f_{\rm NL}$, induce a scale-dependence in the large-scale
bias of halos in the late Universe. This effect is a promising path to
constrain multi-field inflation theories that predict non-zero $f_{\rm NL}$.
While most existing constraints from the halo bias involve auto-correlations of
the galaxy distribution, cross-correlations with probes of the matter density
provide an alternative channel with fewer systematics. We present the strongest
large-scale structure constraint on local primordial non-Gaussianity that uses
cross-correlations alone. We use the cosmic infrared background (CIB)
consisting of dusty galaxies as a halo tracer and cosmic microwave background
(CMB) lensing as a probe of the underlying matter distribution, both from
\textit{Planck} data. Milky Way dust is a key challenge in using the
large-scale modes of the CIB. Importantly, the cross-correlation of the CIB
with CMB lensing is far less affected by Galactic dust compared to the CIB
auto-spectrum, which picks up an additive bias from Galactic dust. We find no
evidence for primordial non-Gaussianity and find $-87<f_{\rm NL}<19$ with a
Gaussian $\sigma(f_{\rm NL})\approx 41$, assuming universality of the halo mass
function. We find that future CMB lensing data from Simons Observatory and
CMB-S4 could achieve $\sigma(f_{\rm NL})$ of 23 and 20 respectively. The
constraining power of such an analysis is limited by current Galactic dust
cleaning techniques, requiring us to use a minimum multipole of $\ell=70$. If
this challenge is overcome with improved analysis techniques or external data,
constraints as tight as $\sigma(f_{\rm NL})=4$ can be achieved through the
cross-correlation technique. More optimistically, constraints better than
$\sigma(f_{\rm NL})=2$ could be achieved if the CIB auto-spectrum is dust-free
down to the largest scales.
