This paper explores the existence of homo- and heteroclinic connections between solar-sail periodic orbits in the planar Earth-Moon circular restricted three-body problem. The existence of such connections has been demonstrated to great extent for the planar and spatial classical (no-solar sail) three-body problem, but remains unexplored for the inclusion of a solar-sail induced acceleration. Similar to the search for homo- and heteroclinic connections in the classical case, this paper uses the tools and techniques of dynamical systems theory, in particular trajectories along the unstable and stable manifolds, to generate these connections. However, due to the time dependency introduced by the solar-sail induced acceleration, common methods and techniques to find homo- and heteroclinic connections (e.g., using the Jacobi constant and applying spatial Poincaré sections) do not necessarily apply. The aim of this paper is therefore to gain an understanding of the extent to which these tools do apply, define new tools (e.g., solar-sail assisted manifolds, temporal Poincaré sections, and a genetic algorithm approach), and ultimately find the sought for homo- and heteroclinic connections. As a starting point of such an investigation, this paper focuses on the planar case, in particular on the search for homo- and heteroclinic connections between three specific solar-sail Lyapunov orbits (two at the L1 point and one at the L2 point) that all exist for the same near-term solar-sail technology. The results of the paper show that, by using a simple solar-sail steering law, where a piece-wise constant sail attitude is applied in the unstable and stable solar-sail manifold trajectories, homo- and heteroclinic connections exist for these three solar-sail Lyapunov orbits. The remaining errors on the position and velocity at linkage of the stable and unstable manifold trajectories are < 10 km and < 1 m/s. Future studies can apply the tools and techniques developed in this paper to extend the search for homo- and heteroclinic connections to other solar-sail Lyapunov orbits in the Earth-Moon system (e.g., for different solar-sail technology), to other planar solar-sail periodic orbits, and ultimately also to the spatial, three-dimensional case.