We report x-ray absorption and x-ray linear dichroism measurements at the Fe L-2,L-3 edges of the geometrically frustrated systems of potassium and hydronium iron jarosite. Comparison with simulated spectra, involving ligand-field multiplet calculations modeling the 3d-2p hybridization between the iron ion and the oxygen ligands, has yielded accurate estimates for the ligand metal-ion hybridization and the resulting single-ion crystal-field anisotropy energy. Using this method we provide an experimentally verified scenario for the appearance of a single-ion anisotropy in this nominally high-spin 3d(5) orbital singlet S-6 system, which accounts for features of the spin-wave dispersion in the long-range-ordered ground state of potassium iron jarosite.