  We compute the general expression of the one-loop vertex correction in an
arbitrary plane-wave background field for the case of two on-shell external
electrons and an off-shell external photon. The properties of the vertex
corrections under gauge transformations of the plane-wave background field and
of the radiation field are studied. Concerning the divergences of the vertex
correction, the infrared one is cured by assigning a finite mass to the photon,
whereas the ultraviolet one is shown to be renormalized exactly as in vacuum.
Finally, the corresponding expression of the vertex correction within the
locally-constant crossed field is also derived and the high-field asymptotic is
shown to scale according to the Ritus-Narozhny conjecture.
