  We provide an existence result for a Schr\"odinger-Poisson system in gradient
form, set in the whole plane, in the case of zero mass. Since the setting is
limiting for the Sobolev embedding, we admit nonlinearities with subcritical or
critical growth in the sense of Trudinger-Moser. In particular, the absence of
the mass term requires a nonstandard functional framework, based on homogeneous
Sobolev spaces. These features, combined with the logarithmic behaviour of the
kernel of the Poisson equation, make the analysis delicate, since standard
variational tools cannot be applied. The system is solved by considering the
corresponding logarithmic Choquard equation. We prove the existence of a
mountain pass-type solution via a careful analysis on specific Cerami
sequences, whose boundedness is achieved by exploiting an appropriate
functional, obtained by evaluating the energy functional on particular paths.
