Real-time vehicle dispatching operations in traditional car-sharing systems is an already computationally challenging scheduling problem. Electrification only exacerbates the computational difficulties as charge level constraints come into play. To overcome this complexity, we employ an online minimum drift plus penalty (MDPP) approach for SAEV systems that (i) does not require a priori knowledge of customer arrival rates to the different parts of the system (i.e. it is practical from a real-world deployment perspective), (ii) ensures the stability of customer waiting times, (iii) ensures that the deviation of dispatch costs from a desirable dispatch cost can be controlled, and (iv) has a computational time-complexity that allows for real-time implementation. Using an agent-based simulator developed for SAEV systems, we test the MDPP approach under two scenarios with real-world calibrated demand and charger distributions: 1) a low-demand scenario with long trips, and 2) a high-demand scenario with short trips. The comparisons with other algorithms under both scenarios show that the proposed online MDPP outperforms all other algorithms in terms of both reduced customer waiting times and vehicle dispatching costs.