The design of online algorithms for matching markets and revenue management settings is usually bound by the assumption that the demand process is formed by a fixed-length sequence of queries with unknown types, each drawn independently. This notion of serial independence implies that the demand of each type, i.e., the number of queries of a given type, has low variance and is approximately Poisson-distributed. This paper proposes a nonparametric framework for modeling arrival sequences in online stochastic matching that departs from the serial independent assumption. We propose two models, Indep and Correl, that capture different forms of serial correlations by combining a nonparametric distribution for the demand with standard assumptions on the arrival patterns -- adversarial or random order. The Indep model can capture arbitrary serial correlations within each customer type but assumes cross-sectional independence across types, whereas the Correl model captures common shocks across customer types. We demonstrate that fluid relaxations, which rely solely on demand expectations, have arbitrarily bad performance guarantees. In contrast, we develop new algorithms that achieve optimal (constant-factor) performance guarantees in each model. Our mathematical analysis includes tighter linear programming (LP) relaxations that leverage distribution knowledge, and a new lossless randomized LP rounding scheme for Indep. We test our new LP relaxations and rounding scheme in simulations on real and synthetic data, and find that they consistently outperform well-established matching algorithms, especially on real data sequences that exhibit greater demand variance.