In this paper, we investigate the convergence rate of the averaging principle for stochastic differential equations (SDEs) with $β$-Hölder drift driven by $α$-stable processes. More specifically, we first derive the Schauder estimate for nonlocal partial differential equations (PDEs) associated with the aforementioned SDEs, within the framework of Besov-Hölder spaces. Then we consider the case where $(α,β)\in(0,2)\times(1-\tfracα{2},1)$. Using the Schauder estimate, we establish the strong convergence rate for the averaging principle. In particular, under suitable conditions we obtain the optimal rate of strong convergence when $(α,β)\in(\tfrac{2}{3},1]\times(2-\tfrac{3α}{2},1)\cup(1,2)\times(\tfracα{2},1)$. Furthermore, when $(α,β)\in(0,1]\times(1-α,1-\tfracα{2}]\cup(1,2)\times(\tfrac{1-α}{2},1-\tfracα{2}]$, we show the convergence of the martingale solutions of original systems to that of the averaged equation. When $α\in(1,2)$, the drift can be a distribution.