As an integral component of space-air-ground integrated networks (SAGINs), the low Earth orbit (LEO) satellite networks have displayed immense potential in providing ubiquitous connectivity and broadband mobile communication. However, the intrinsic dynamics of LEO satellites poses unprecedented challenges in network management, multi-dimensional resource scheduling, and service delivery. In this paper, we study the service function chain (SFC) orchestration in dynamic LEO satellite networks, with the aim of achieving flexible and efficient service provision. Considering the service requirements and the load fairness of LEO satellite networks, we formulate the SFC deployment problem as an integer nonlinear programming (INLP) problem. We then introduce a load-aware SFC orchestration algorithm to improve serving capacity and load fairness. Additionally, we address the issue of SFC migration in dynamic LEO satellite networks to ensure service continuity. To minimize the service interruption and network resource wastes, a Tabu search (TS)-based approach is presented to optimize the virtual network function (VNF) migration. Simulation results demonstrate that our proposed approaches outperform the benchmark by a substantial margin in terms of load fairness, without compromising service acceptance.