The prospects of extracting new physics signals in a coherent elastic neutrino-nucleus scattering (CEνNS) process are limited by the precision with which the underlying nuclear structure physics, embedded in the weak nuclear form factor, is known. We present microscopic nuclear structure physics calculations of charge and weak nuclear form factors and CEνNS cross sections on 12 C, 16 O, 40 Ar, 56 Fe and 208 Pb nuclei. We obtain the proton and neutron densities, and charge and weak form factors by solving Hartree-Fock equations with a Skyrme (SkE2) nuclear potential. We validate our approach by comparing 208 Pb and 40 Ar charge form factor predictions with elastic electron scattering data. In view of the worldwide interest in liquid-argon based neutrino and dark matter experiments, we pay special attention to the 40 Ar nucleus and make predictions for the 40 Ar weak form factor and the CEνNS cross sections. Furthermore, we attempt to gauge the level of theoretical uncertainty pertaining to the description of the 40 Ar form factor and CEνNS cross sections by comparing relative differences between recent microscopic nuclear theory and widely-used phenomenological form factor predictions. Future precision measurements of CEνNS will potentially help in constraining these nuclear structure details that will in turn improve prospects of extracting new physics.