We present the proton and neutron vector form factors in a convenient parametric form that is model independent and optimized for momentum transfers $\leq$ few GeV$^2$. The form factors are determined from a global fit to electron scattering data and precise charge radii measurements. We apply a new treatment of radiative corrections. We evaluate the neutrino–nucleon scattering cross sections at GeV energies of neutrino oscillation experiments. The neutrino–nucleon charged current quasielastic cross section differs by 3–5% compared to commonly-used form factor models when the vector form factors are constrained by recent high-statistics electron-proton scattering data from the A1 collaboration. A primary goal of this work is to provide a consistent framework for applications such as neutrino event generators to propagate form factor constraints and uncertainties into cross section predictions.