As one representative of last-mile logistics in intelligent transportation systems, the on-demand food delivery (OFD) service has gained rapid market growth but also faces multiple challenges. One of the critical issues is the order dispatching problem (ODP) with an NP-hard nature, which refers to dispatching a large number of orders to riders reasonably in real time with very limited decision time. To address the ODP, this paper proposes an optimization algorithm based on graph neural networks (GNN) by combining the advantages of machine learning (ML) techniques and operational research (OR) methods: 1) The ML component learns to reduce the solution space by filtering out inappropriate riders for each order, handling the large-scale complexity of ODP. Specifically, we present a rider modeling approach by using GNN to better characterize rider information; besides, two attention mechanisms are designed to adaptively learn the matching relationship between riders and orders. 2) The OR component ensures the solution quality with a greedy and regret value-based dispatching heuristic. Extensive experiments are conducted on real-world datasets to evaluate the performance of the proposed method by comparing it with other existing models and algorithms. The results show that the design of our ML model is effective in yielding better prediction results, and the proposed GNN-based optimization algorithm can effectively and efficiently solve the ODP by improving delivery efficiency and customer satisfaction.