In the non-perturbative setting of algebraic QFT we give a mathematically rigorous construction of the scattering matrix for massive Wigner particles in presence of massless excitations. In contrast to previous approaches we do not impose any Herbst-type technical assumptions on the spectrum of the mass operator near the particle masses. Instead we base our approach on features of the relativistic vacuum state which are similar to the well-established Reeh-Schlieder property. The method should apply, in particular, to scattering of stable particles in abelian gauge theories. A concrete example are hydrogen atoms from the point of view of Quantum Electrodynamics (based on CMP 352, 935-966, 2017).