We explicitly account for electron-electron interactions when modeling low-dimensional helical organic molecules. We show that competition between various hopping channels, together with interaction-induced double- and superexchange mechanisms, can stabilize non-collinear helical magnetic order. The resulting single-electron bands exhibit partial spin polarization, a manifestation of $p$-wave magnetism. Using density-matrix renormalization group, cluster perturbation theory, and Monte Carlo methods, we find that even vanishingly small spin-orbit coupling triggers strong spin selectivity at temperatures significantly above the spin-orbit scale. While strong correlations are essential for this mechanism, long-range spin ordering is not required. We thus propose non-collinear spin correlations driven by Coulomb interactions as an explanation of chirality-induced spin selectivity and discuss connections to experiments.