By establishing the regularity estimates for nonlocal Stein/Poisson equations under $γ$-order Hölder and dissipative conditions on the coefficients, we derive the $W_{\bf d}$-convergence rate for the Euler-Maruyama schemes applied to the invariant measure of SDEs driven by multiplicative $α$-stable noises with $α\in (\frac{1}{2}, 2)$, where $W_{\bf d}$ denotes the Wasserstein metric with ${\bf d}(x,y)=|x-y|^γ\wedge 1$ and $γ\in ((1-α)_+, 1]$.