We study a federated classification problem over a network of multiple clients and a central server, in which each client's local data remains private and is subject to uncertainty in both the features and labels. To address these uncertainties, we develop a novel Federated Distributionally Robust Support Vector Machine (FDR-SVM), robustifying the classification boundary against perturbations in local data distributions. Specifically, the data at each client is governed by a unique true distribution that is unknown. To handle this heterogeneity, we develop a novel Mixture of Wasserstein Balls (MoWB) ambiguity set, naturally extending the classical Wasserstein ball to the federated setting. We then establish theoretical guarantees for our proposed MoWB, deriving an out-of-sample performance bound and showing that its design preserves the separability of the FDR-SVM optimization problem. Next, we rigorously derive two algorithms that solve the FDR-SVM problem and analyze their convergence behavior as well as their worst-case time complexity. We evaluate our algorithms on industrial data and various UCI datasets, whereby we demonstrate that they frequently outperform existing state-of-the-art approaches.