Multi-access Edge Computing (MEC) delivers low-latency services by hosting applications near end-users. To promote sustainability, these systems are increasingly integrated with renewable Energy Harvesting (EH) technologies, enabling operation where grid electricity is unavailable. However, balancing the intermittent nature of harvested energy with dynamic user demand presents a significant resource allocation challenge. This work proposes an online strategy for an MEC system powered exclusively by EH to address this trade-off. Our strategy dynamically schedules computational tasks with dependencies and governs energy consumption through real-time decisions on server frequency scaling and service module migration. Experiments using real-world datasets demonstrate our algorithm's effectiveness in efficiently utilizing harvested energy while maintaining low service latency.