As an integral component of the space-air-ground integrated network (SAGIN), the low Earth orbit (LEO) satellite network has displayed immense potential in providing ubiquitous connectivity and broadband mobile communication. However, the intrinsic dynamics of LEO satellites pose unprecedented challenges in network management and service delivery. In this paper, we investigate the service function chain (SFC) orchestration in dynamic LEO satellite networks to achieve flexible and efficient service provision. Considering the service requirements and the limitations of network resources, we formulate the SFC orchestration problem as the integer nonlinear programming (INLP) problem for maximizing the service acceptance and the load fairness of satellites. Then, an efficient heuristic algorithm is proposed to solve this problem. Addressing the situation with frequent service requests, a graph attention network (GAT)-based approach with low complexity is also presented. Simulation results demonstrate that our proposed approaches outperform the benchmarks by a substantial margin in terms of load fairness and service acceptance. Besides, the proposed GAT-based approach shows its advantage in computation complexity, and exhibits robustness in unstable network scenarios with intermittent link interruptions.