  We consider the dispersion on the supernova distance-redshift relation due to
peculiar velocities and gravitational lensing, and the sensitivity of these
effects to the amplitude of the matter power spectrum. We use the MeMo lensing
likelihood developed by Quartin, Marra & Amendola (2014), which accounts for
the characteristic non-Gaussian distribution caused by lensing magnification
with measurements of the first four central moments of the distribution of
magnitudes. We build on the MeMo likelihood by including the effects of
peculiar velocities directly into the model for the moments. In order to
measure the moments from sparse numbers of supernovae, we take a new approach
using Kernel Density Estimation to estimate the underlying probability density
function of the magnitude residuals. We also describe a bootstrap re-sampling
approach to estimate the data covariance matrix. We then apply the method to
the Joint Light-curve Analysis (JLA) supernova catalogue. When we impose only
that the intrinsic dispersion in magnitudes is independent of redshift, we find
$\sigma_8=0.44^{+0.63}_{-0.44}$ at the one standard deviation level, although
we note that in tests on simulations, this model tends to overestimate the
magnitude of the intrinsic dispersion, and underestimate $\sigma_8$. We note
that the degeneracy between intrinsic dispersion and the effects of $\sigma_8$
is more pronounced when lensing and velocity effects are considered
simultaneously, due to a cancellation of redshift dependence when both effects
are included. Keeping the model of the intrinsic dispersion fixed as a Gaussian
distribution of width 0.14 mag, we find $\sigma_8 = 1.07^{+0.50}_{-0.76}$.
