Motivated by the omnipresence of extreme value distributions in limit theorems involving extremes of random processes, we adapt Stein's method to include these laws as possible target distributions. We do so by using the generator approach of Stein's method, which is possible thanks to a recently introduced family of semi-groups. We study the corresponding Stein solution and its properties when the working distance is either the smooth Wasserstein distance or the Kolmogorov distance. We make use of those results to bound the distance between two max-stable random vectors, as well as to get a rate of convergence for the de Haan-LePage series in smooth Wasserstein distance.