Space-air-ground integrated networks (SAGINs) are envisioned to provide seamless coverage and enhanced flexibility compared with traditional terrestrial mobile networks, which has attracted much attention from both industry and academia. However, orchestrating heterogeneous resources in such a large-scale and dynamic network is challenging, especially encountering diverse services with multi-dimensional requirements. In this paper, we first propose a software-defined networking (SDN) and network function virtualization (NFV)-based reconfigurable SAGIN architecture for constructing service function chains (SFCs). Based on that, we investigate the SFC orchestration and wireless resource management where the virtual link rate adaption between each virtual network function (VNF) is introduced to improve the network resource utilization. Considering the limited physical resource and the heterogeneity in SAGINs, we jointly formulate the VNF embedding, virtual link rate adaption, and wireless resource allocation as a mixed-integer nonlinear programming (MINLP) problem to maximize the network profit. Due to the NP-hardness of the problem, we first transform the problem into a continuous optimization problem by successive convex approximation. By introducing an additional penalty into the objective function, an iterative alternation algorithm is proposed to find a near-optimal solution of the transformed problem. Extensive simulation results show that our proposed approach outperforms the benchmarks in average network revenue, successfully serving probability, and resource consumption.