We consider a family of polynomial analogs of the Apéry numbers, which includes $q$-analogs of Krattenthaler--Rivoal--Zudilin and Zheng, and show that the supercongruences that Gessel and Mimura established for the Apéry numbers generalize to these polynomials. Our proof relies on polynomial analogs of classical binomial congruences of Wolstenholme and Ljunggren. We further indicate that this approach generalizes to other supercongruence results.