Autonomous vehicles (AVs) have the potential to enhance road safety, reduce fuel consumption, and alleviate traffic congestion. However, the computational demands of processing large volumes of sensory data for tasks like motion planning and trajectory forecasting impose critical challenges on limited onboard resources of AVs. Mobile edge computing (MEC) offers a solution by offloading these tasks to edge servers located in prox-imity of the vehicles. When AVs traverse remote areas lacking terrestrial infrastructures, low Earth orbit (LEO) satellites can fill this gap by providing edge computing services. In this paper, we propose a microservice-based framework for MEC-enabled hybrid satellite-terrestrial networks to support AVs. By decomposing monolithic applications into microservices deployed in containers, we enable scalable and flexible computing services. We address the challenges of microservice migration due to the mobility of AVs and LEO satellites by formulating a long-term optimization problem aimed at minimizing task and migration delays. An online Lyapunov-based algorithm is developed to solve this problem, reducing the decision space by scheduling periodic migrations and decomposing it into mixed-integer linear program-ming. Numerical results demonstrate that our proposed algorithm can achieve a nearly optimal results while maintaining a low execution time.