New monoclinic (P2/c) tungstates – a medium-entropy tungstate, (MnNiCuZn)WO4, and a high-entropy tungstate, (MnCoNiCuZn)WO4 – were synthesized and characterized. Their phase purity and solid solution nature were confirmed by powder X-ray diffraction and Raman spectroscopy. X-ray absorption spectroscopy was used to probe the local structure around metal cations. The atomic structures based on the ideal solid solution model were optimized by a simultaneous analysis of the extended X-ray absorption fine structure spectra at multiple metal absorption edges – five for (MnNiCuZn)WO4 and six for (MnCoNiCuZn)WO4 – by means of reverse Monte Carlo simulations. In both compounds, Ni2+ ions have the strongest tendency to organize their local environment and form slightly distorted [NiO6] octahedra, whereas Mn2+, Co2+, and Zn2+ ions have a strongly distorted octahedral coordination. The most intriguing result is that the shape of [CuO6] octahedra in (MnNiCuZn)WO4 and (MnCoNiCuZn)WO4 differs from that found in pure CuWO4, where a strong Jahn–Teller distortion is present: [CuO6] octahedra become more regular with increasing degree of dilution.