  We introduce a non-perturbative method to constrain the amplitude of
local-type primordial non-Gaussianity ($f_{\rm NL}$) using squeezed
configurations of the CMB lensing convergence and cosmic shear bispectra.
First, we use cosmological consistency relations to derive a model for the
squeezed limit of angular auto- and cross-bispectra of lensing convergence
fields in the presence of $f_{\rm NL}$. Using this model, we perform a Fisher
forecast with specifications expected for upcoming CMB lensing measurements
from the Simons Observatory and CMB-S4, as well as cosmic shear measurements
from a Rubin LSST/Euclid-like experiment. Assuming a minimum multipole
$\ell_{\rm min}=10$ and maximum multipole $\ell_{\rm max}=1400$, we forecast
$\sigma_{f_{\rm NL}}=175$ ($95$) for Simons Observatory (CMB-S4). Our forecasts
improve considerably for an LSST/Euclid-like cosmic shear experiment with three
tomographic bins and $\ell_{\rm min}=10$ and $\ell_{\rm max}=1400$ ($5000$)
with $\sigma_{f_{\rm NL}}=31$ ($16$). A joint analysis of CMB-S4 lensing and
LSST/Euclid-like shear yields little gain over the shear-only forecasts;
however, we show that a joint analysis could be useful if the CMB lensing
convergence can be reliably reconstructed at larger angular scales than the
shear field. The method presented in this work is a novel and robust technique
to constrain local primordial non-Gaussianity from upcoming large-scale
structure surveys that is completely independent of the galaxy field (and
therefore any nuisance parameters such as $b_\phi$), thus complementing
existing techniques to constrain $f_{\rm NL}$ using the scale-dependent halo
bias.
