  Self-interacting dark matter (SIDM) is an able alternative to collisionless
dark matter. If dark matter does have self-interactions, we would expect this
to cause a separation between the collisionless stars and the dark matter halo
of a galaxy as it falls through a dark matter medium. For stars arranged in a
disk, this would generate a U-shaped warp. The magnitude of this warping
depends on the SIDM cross section, type of self-interaction, relative velocity
of galaxy and background, halo structure, and density of the dark matter
medium. In this paper, we set constraints on long-range (light mediator) dark
matter self-interaction by means of this signal. We begin by measuring U-shaped
warps in $3,213$ edge-on disk galaxies within the Sloan Digital Sky Survey. We
then forward-model the expected warp from SIDM on a galaxy-by-galaxy basis by
combining models of halo structure, density and velocity field reconstructions,
and models for the dark matter interactions. We find no evidence for a
contribution to the warps from SIDM. Our constraints are highly dependent on
the uncertain velocities of our galaxies: for a normalized Rutherford-like
cross section we find $\tilde{\sigma}/m_{\rm{DM}} \lesssim 3\times
10^{-13}~\rm{cm}^2/\rm{g}$ at fixed velocity $v = 300~\rm{km/s}$ -- a bound
that scales roughly linearly with increasing $v$. In the appendix we translate
these bounds into limits on the momentum transfer cross section, finding
$\sigma_T(300~\rm{km/s})/m_{\rm{DM}} \lesssim 0.1~\rm{cm}^2/\rm{g}$. [abridged]
