The influence of a gate voltage on magnetic anisotropy is investigated in a thin Fe film epitaxially grown on a Ag(1,1,10) substrate and covered by MgO. Oscillations in step-induced magnetic anisotropy due to quantum well states (QWS) confined in the Fe film are observed and shown to persist up to room temperature at low Fe thicknesses. By systematically examining the voltage and thickness dependence of the magnetic hysteresis loop characteristics, we identify two distinct effects by which an applied voltage modifies the magnetic anisotropy. The first effect is due to voltage-induced changes to interfacial perpendicular magnetic anisotropy which, due to the vicinal geometry, leads to changes in the effective in-plane uniaxial magnetic anisotropy. A second effect is observed at lower film thicknesses and shows nonmonotonic voltage-induced effects on magnetic anisotropy. This nonmonotonic behavior coincides with the onset of significant QWS-induced effects on magnetic anisotropy and suggests a link between QWS- and voltage-induced anisotropy changes.