The orbital contribution to the magnetic moment of the transition-metal ion in the isostructural weak ferromagnets ACO(3) (A = Mn,Co,Ni) and FeBO3 was investigated by a combination of first-principles calculations, nonresonant x-ray magnetic scattering, and x-ray magnetic circular dichroism. A nontrivial evolution of the orbital moment as a function of the 3d orbitals filling is revealed, with a particularly large value found in the Co member of the family. Here, the coupling between magnetic and lattice degrees of freedom produced by the spin-orbit interaction results in a large single-ion anisotropy and a peculiar magnetic-moment-induced electron cloud distortion, evidenced by the appearance of a subtle scattering amplitude at space-group-forbidden reflections and significant magnetostrictive effects. Our results, which complement a previous investigation on the sign of the Dzyaloshinskii-Moriya interaction across the series, highlight the importance of spin-orbit coupling in the physics of weak ferromagnets and prove the ability of modern first-principles calculations to predict the properties of materials where the Dzyaloshinskii-Moriya interaction is a fundamental ingredient of the magnetic Hamiltonian.