We present a variation of the electric-field volume integral equation (VIE) for inhomogeneous media that is tailored for microwave S-parameter measurements. The kernel of the standard VIE is a dyadic Green's function that can relate fields in the object domain to observed scattered fields outside the object domain. Here, we derive a kernel we call a vector Green's function that gives, instead of observed fields, observed S-parameter measurements that would be obtained in a bistatic antenna setup. This formulation directly links the object properties to microwave measurements in a way that is useful to many microwave applications. This derivation uses an antenna model based on the source-scattering matrix formulation. We show how this vector Green's function can be obtained from simulation and validate this model with several experiments.