  Upcoming surveys such as \LSST{} and \Euclid{} will significantly improve the
power of weak lensing as a cosmological probe. To maximise the information that
can be extracted from these surveys, it is important to explore novel
statistics that complement standard weak lensing statistics such as the
shear-shear correlation function and peak counts. In this work, we use a
recently proposed weak lensing observable -- weak lensing voids -- to make
parameter constraint forecasts for an LSST-like survey. We use the
\cosmoslics{} $w$CDM simulation suite to measure void statistics as a function
of cosmological parameters. The simulation data is used to train a Gaussian
process regression emulator that we use to generate likelihood contours and
provide parameter constraints from mock observations. We find that the void
abundance is more constraining than the tangential shear profiles, though the
combination of the two gives additional constraining power. We forecast that
without tomographic decomposition, these void statistics can constrain the
matter fluctuation amplitude, $S_8$ within 0.3\% (68\% confidence interval),
while offering 1.5, 1.5 and 2.7\% precision on the matter density parameter,
$\Omega_{\rm m}$, the reduced Hubble constant, $h$, and the dark energy
equation of state parameter, $w_0$, respectively. These results are tighter
than the constraints from the shear-shear correlation function with the same
observational specifications for $\Omega_m$, $S_8$ and $w_0$. The constraints
from the WL voids also have complementary parameter degeneracy directions to
the shear 2PCF for all combinations of parameters that include $h$, making weak
lensing void statistics a promising cosmological probe.
