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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.