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The interdependence between electric power systems and transportation systems is rapidly increasing due to the high adoption of Electric Vehicles (EVs) and their charging infrastructures. Electric vehicles can represent additional load for the power system, but can also bring new opportunities for contributing to the efficient and resilient operations of the power grid. This is mainly because of their ability to provide back power to the system when it is not used for transportation, essentially serving as a moving battery source for the power grid. This so-called Vehicle-to-Grid (V2G), Vehicle-to-Building (V2B), or, generally, Vehicle-to-X (V2X) capability of EVs has been extensively studied in the literature. However, the upcoming development of autonomous driving systems and their integration within sharing mobility services can significantly add to the possibilities of interaction between the transportation system and the power grid. This paradigm is studied to a much lesser extent in the existing literature. Shared Autonomous Electric Vehicles (SAEVs) could allow for more control of the actions of the fleet allowing for large-scale coordinated responses both to mobility and energy demands. This coordinated response can be particularly useful for providing emergency power services in case of power loss while maintaining a high level of transportation service. Thus, improving the overall resilience of the system. In this work, we develop a dynamic optimization framework to evaluate the potential contribution of the SAEV fleet for improving critical buildings' energy resilience via V2B services. The model considers passengers' pick-up and transportation, relocation of vehicles, and battery charging and discharging. Power outage scenarios for critical buildings are considered and the potential of the SAEV fleet to fully or partially respond to the emergency power outage is studied. In addition, sensitivity analysis for key parameters such as the outage parameters is introduced. The results of the case study for the Ile-de-France region in France shows that the SAEV fleet has the potential to provide V2B service for critical building at an acceptable loss of passenger total waiting time on the transportation side. Furthermore, it is shown that it is capable of satisfying the emergency power load at a lower cost compared to investing in extra backup generators unless the outage occurs at significantly high frequencie