The relativistic vortex states of massive charged particles propagating in non-uniform axisymmetric electric field are studied. Starting from the stationary-state equation after the relativistic Foldy–Wouthuysen (FW) transformation and employing the Scherzer expansion, a longitudinal WKB reduction, and the paraxial approximation, the coupled evolution equations for the beam width, wavefront curvature, and Gouy phase are derived. The equations are solved numerically for a quadratic electrostatic potential, an immersion lens, and an einzel lens. The essential influence of the transverse field on beam evolution is demonstrated. The results offer insights into the evolution of accelerated vortex electron beams and may help guide the control of their transverse properties using electrostatic fields.