This paper assesses the equity impacts of for-hire autonomous vehicles (AVs) and investigates regulatory policies that promote the spatial and social equity in future autonomous mobility ecosystems. To this end, we consider a multimodal transportation network, where a ride-hailing platform operates a fleet of AVs to offer mobility-on-demand services in competition with a public transit agency that offers transit services on a transportation network. A game-theoretic model is developed to characterize the intimate interactions between the ride-hailing platform, the transit agency, and multiclass passengers with distinct income levels. An algorithm is proposed to compute the Nash equilibrium of the game and conduct an ex-post evaluation of the performance of the obtained solution. Based on the proposed framework, we evaluate the spatial and social equity in transport accessibility using Theil index, and find that although the proliferation of for-hire AVs in the ride-hailing network improves overall accessibility, the benefits are not fairly distributed among distinct locations or population groups, implying that the deployment of AVs will enlarge the existing spatial and social inequity gaps in the transportation network if no regulatory intervention is in place. To address this concern, we investigate two regulatory policies that can improve transport equity: (a) a minimum service-level requirement on ride-hailing service, which improves the spatial equity in the transport network; (b) a subsidy on ride-hailing trips that serve as first/last-mile connection to public transit, which promotes the use of public transit and improves the social equity of the transport network. We show that the minimum service-level requirement entails a trade-off: as a higher minimum service level is imposed, the spatial inequity reduces, but the social inequity will be exacerbated. In contrast, subsidies on bundled trips can enhance accessibility, increase transit ridership, improve the profit of the ride-hailing platform, and more crucially, simultaneously bridge spatial and social inequity gaps. These results are validated through realistic numerical studies for San Francisco.