  Core formation and runaway core collapse in models with self-interacting dark
matter (SIDM) significantly alter the central density profiles of collapsed
halos. Using a forward modeling inference framework with simulated datasets, we
demonstrate that flux ratios in quadruple image strong gravitational lenses can
detect the unique structural properties of SIDM halos, and statistically
constrain the amplitude and velocity dependence of the interaction cross
section in halos with masses between $10^6 - 10^{10} M_{\odot}$. Measurements
on these scales probe self-interactions at velocities below $30 \ \rm{km} \
\rm{s^{-1}}$, a relatively unexplored regime of parameter space, complimenting
constraints at higher velocities from galaxies and clusters. We cast
constraints on the amplitude and velocity dependence of the interaction cross
section in terms of $\sigma_{20}$, the cross section amplitude at $20 \ \rm{km}
\ \rm{s^{-1}}$. With 50 lenses, a sample size available in the near future, and
flux ratios measured from spatially compact mid-IR emission around the
background quasar, we forecast $\sigma_{20} < 11-23 \ \rm{cm^2} \rm{g^{-1}}$ at
$95 \%$ CI, depending on the amplitude of the subhalo mass function, and
assuming cold dark matter (CDM). Alternatively, if $\sigma_{20} = 19.2 \
\rm{cm^2}\rm{g^{-1}}$ we can rule out CDM with a likelihood ratio of 20:1,
assuming an amplitude of the subhalo mass function that results from
doubly-efficient tidal disruption in the Milky Way relative to massive
elliptical galaxies. These results demonstrate that strong lensing of compact,
unresolved sources can constrain SIDM structure on sub-galactic scales across
cosmological distances, and the evolution of SIDM density profiles over several
Gyr of cosmic time.
