We apply the scattering approach to the Casimir interaction between two dielectric half-spaces separated by an electrolyte solution. We take the nonlocal electromagnetic response of the intervening medium into account, which results from the presence of movable ions in solution. In addition to the usual transverse modes, we consider longitudinal channels and their coupling by reflection at the surface of the local dielectric. The Casimir interaction energy is calculated from the matrix describing a round-trip of coupled transverse and longitudinal waves between the interacting surfaces. The nonzero-frequency contributions are approximately unaffected by the presence of ions. We find, at zero frequency, a contribution from longitudinal channels, which is screened over a distance of the order of the Debye length, alongside an unscreened term arising from transverse-magnetic modes. The latter defines the long-distance asymptotic limit for the interaction.