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As electric vehicle (EV) technologies become mature, EV has been rapidly adopted in modern transportation systems, and is expected to provide future autonomous mobility-on-demand (AMoD) service with economic and societal benefits. However, EVs require frequent recharges due to their limited and unpredictable cruising ranges, and they have to be managed efficiently given the dynamic charging process. It is urgent and challenging to investigate a computationally efficient algorithm that provide EV AMoD system performance guarantees under model uncertainties, instead of using heuristic demand or charging models. To accomplish this goal, this work designs a data-driven distributionally robust optimization approach for vehicle supply-demand ratio and charging station utilization balancing, while minimizing the worst-case expected cost considering both passenger mobility demand uncertainties and EV supply uncertainties. We then derive an equivalent computationally tractable form for solving the distributionally robust problem in a computationally efficient way under ellipsoid uncertainty sets constructed from data. Based on E-taxi system data of Shenzhen city, we show that the average total balancing cost is reduced by 14.49%, the average unfairness of supply-demand ratio and utilization is reduced by 15.78% and 34.51% respectively with the distributionally robust vehicle balancing method, compared with solutions which do not consider model uncertainties.