Microwave spectra are reported for the C3v symmetric complexes Kr-SO3 and Kr-SO3-CO. The S-C distance in the trimer, 2.871(9) Å, is the same as that previously determined for SO3-CO to within the estimated uncertainties. The Kr-S distances are 3.438(3) Å and 3.488(6) Å in Kr-SO3 and Kr-SO3-CO, respectively, indicating that the addition of CO to Kr-SO3 increases the Kr-S distance by 0.050(9) Å. Measurements of the (83)Kr nuclear quadrupole coupling constants provide direct probes of the electric field gradient at the Kr nucleus, and a comparison between the two systems reflects the degree to which the CO influences the electronic structure of the krypton atom. Although the Kr and CO in the trimer are on opposite sides of the SO3 and thus are not in direct contact, the addition of CO to Kr-SO3 reduces the electric field gradient at the Kr nucleus by 18%. Calculations using the block localized wavefunction decomposition method are performed to understand the physical origins of this change. While the magnitudes of both the electric field and the electric field gradient at the Kr nucleus decrease upon addition of the CO to Kr-SO3, the changes are shown to arise from rather complex combinations of geometrical distortion, electrostatic, polarization, and electron transfer effects. For the electric field, the electrostatic term accounts for the largest portion of the reduction, while for the electric field gradient, polarization and structural change of the Kr-SO3 moiety make the primary contributions. Despite significant changes in the electronic environment at the Kr nucleus, calculated binding energies indicate that the interactions are largely additive, with the binding energy of the trimer very nearly equal to the sum of the Kr-SO3 and SO3-CO binding energies.