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Shared-use autonomous mobility services (SAMS) present new opportunities for improving accessible and demand-responsive mobility. A fundamental challenge that SAMS face is appropriate positioning of idle fleet vehicles to meet future demand - a problem that strongly impacts service quality and efficiency. This paper formulates SAMS fleet repositioning as a Markov Decision Process and presents a reinforcement learning-based repositioning (RLR) approach called integrated system-agent repositioning (ISR). The ISR learns a scalable fleet repositioning strategy in an integrated manner: learning to respond to evolving demand patterns without explicit demand forecasting and to cooperate with optimization-based passenger-to-vehicle assignment. Numerical experiments are conducted using New York City taxi data and an agent-based simulation tool. The ISR is compared to an alternative RLR approach named externally guided repositioning (EGR) and a benchmark joint optimization (JO) for passenger-to-vehicle assignment and repositioning. The results demonstrate the RLR approaches' substantial reductions in passenger wait times, over 50%, relative to the JO approach. The ISR's ability to bypass demand forecasting is also demonstrated as it maintains comparable performance to EGR in terms of average metrics. The results also demonstrate the model's transferability to evolving conditions, including unseen demand patterns, extended operational periods, and changes in the assignment strategy.