We present the nucleon vector form factors in a convenient parametric form that is optimized for low momentum transfers < few GeV^2. The form factors are determined from a global fit to electron scattering data and precise charge radii measurements with an updated treatment of radiative corrections. We evaluate the unpolarized charged-current elastic neutrino–nucleon scattering cross sections at GeV energies of neutrino oscillation experiments. These cross sections differ by 3–5% compared to commonly used form-factor models. Contrary to these models, our form factors are constrained by recent high-statistics electron–proton scattering data from the A1 Collaboration. We also implement updated nucleon vector form factors in the neutrino event generator GENIE. On top of the form-factor description at leading order, radiative corrections in the Standard Model and potential new physics generate additional contributions. We provide a general framework of invariant amplitudes and explore the impact of modern and future cross section measurements, considering both unpolarized cross sections and polarization observables, on constraining these amplitudes. We also discuss the effects of radiative corrections on the unpolarized cross section and all possible single-spin asymmetries.