We price American options using kernel-based approximations of the Volterra Heston model. We choose these approximations because they allow simulation-based techniques for pricing. We prove the convergence of American option prices in the approximating sequence of models towards the prices in the Volterra Heston model. A crucial step in the proof is to exploit the affine structure of the model in order to establish explicit formulas and convergence results for the conditional Fourier--Laplace transform of the log price and an adjusted version of the forward variance. We illustrate with numerical examples our convergence result and the behavior of American option prices with respect to certain parameters of the model.