Recent theoretical work has solidified the viability of the collisional runaway scenario in young dense star clusters for the formation of very massive stars (VMSs), which may be precursors to intermediate-mass black holes (IMBHs). We present first results from a numerical study of the collisional runaway process in dense star clusters containing primordial binaries. Stellar collisions during binary scattering encounters provide an alternate channel for runaway growth, somewhat independent of direct collisions between single stars. We find that clusters with binary fractions greater than or similar to 10% yield two VMSs via collisional runaways, presenting the exotic possibility of forming IMBH-IMBH binaries in star clusters. We discuss the implications for gravitational wave observations and the impact on cluster structure.