As China continues to develop a unified national gas pipeline network, operators must coordinate pipeline flows with underground gas storage (UGS) injection and withdrawal to ensure peak-period gas supply. To address heterogeneous seasonal demand patterns across user types and deviations of realized demand from forecasts, this study develops a distributionally robust optimization framework for integrated natural gas transmission and storage operations under demand uncertainty. First, FEDformer is used to forecast monthly demand for city-gas, industrial, power-generation, and chemical users. Historical relative forecast residuals are then resampled using the Bootstrap method and clustered to construct an empirical distribution of future demand. Second, a Wasserstein ambiguity set centered on this distribution is used to formulate a multi-period distributionally robust optimization model that jointly determines pipeline flows and UGS injection, withdrawal, and inventory plans. Finally, the model is reformulated as a mixed-integer linear program through Lagrangian duality and solved using Gurobi. Numerical experiments show that FEDformer achieves an average MAPE of 3.58%, 10.74–54.03% lower than that of nine benchmark models. Across five supply–demand scenarios, the proposed model achieves the lowest or tied-lowest unmet demand, while reducing total cost by CNY 40–242 million and unmet demand by 64.58–100% relative to deterministic optimization. Under demand-fluctuation scenarios, it also yields lower total costs than stochastic, conventional robust, and interval robust optimization.