We study distributionally robust chance constrained programs (DRCCP) with maximum mean discrepancy (MMD) ambiguity sets. We provide an exact reformulation of those problems such that the uncertain constraint is satisfied with a probability larger than a desired risk-level for distributions within the MMD ball around the empirical distribution. Additionally, we highlight how the ambiguity set can be connected to known statistical bounds on the MMD to obtain statistical guarantees for the data-driven DRCCP. Lastly, we validate our reformulation on a numerical example and compare it to the robust scenario approach.