In this paper, we study a distributionally robust optimization approach to chance-constrained stochastic programs to hedge against uncertainty in the distributions of the random parameters. We consider a general polyhedral ambiguity set under finite support and study Wasserstein ambiguity set, total variation distance ambiguity set, and moment-based ambiguity set as examples for our computations. We develop a decomposition-based solution approach to solve the model and take advantage of mixing inequalities to develop custom feasibility cuts. A probability cut framework is also developed to handle the distributionally robust chance constraint. Finally, we present a numerical study to illustrate the effectiveness of the proposed formulations and showcase our results for the chosen ambiguity set examples.