  We explore the impact of elastic, anisotropic, velocity-dependent dark matter
(DM) self-interactions on the host halo and subhalos of Milky Way (MW)--mass
systems. We consider a generic self-interacting dark matter (SIDM) model
parameterized by the masses of a light mediator and the DM particle. The ratio
of these masses, $w$, sets the velocity scale above which momentum transfer due
to DM self-interactions becomes inefficient. We perform high-resolution zoom-in
simulations of an MW-mass halo for values of $w$ that span scenarios in which
self-interactions either between the host and its subhalos or only within
subhalos efficiently transfer momentum, and we study the effects of
self-interactions on the host halo and on the abundance, radial distribution,
orbital dynamics, and density profiles of subhalos in each case. The abundance
and properties of surviving subhalos are consistent with being determined
primarily by subhalo--host halo interactions. In particular, subhalos on radial
orbits in models with larger values of the cross section at the host halo
velocity scale are more susceptible to tidal disruption owing to mass loss from
ram pressure stripping caused by self-interactions with the host. This
mechanism suppresses the abundance of surviving subhalos relative to
collisionless DM simulations, with stronger suppression for larger values of
$w$. Thus, probes of subhalo abundance around MW-mass hosts can be used to
place upper limits on the self-interaction cross section at velocity scales of
$\sim 200\ \rm{km\ s}^{-1}$, and combining these measurements with the orbital
properties and internal dynamics of subhalos may break degeneracies among
velocity-dependent SIDM models.
