Public libraries are an essential public good. We ask: are urban library systems providing equitable service to all residents, in terms of the books they have access to and check out? If not, what causes disparities: heterogeneous book collections, resident behavior and access, and/or operational policies? Existing methods leverage only system-level outcome data (such as overall checkouts per branch), and so cannot distinguish between these factors. As a result, it is difficult to use their results to guide interventions to increase equitable access. We propose a Bayesian framework to characterize book checkout behavior across multiple branches of a library system, learning heterogeneous book popularity, overall branch demand, and usage of the online hold system, while controlling for book availability. In collaboration with the New York Public Library, we apply our framework to granular data consisting of over 400,000 checkouts during 2022. We first show that our model significantly out-performs baseline methods in predicting checkouts at the book-branch level. Next, we study spatial and socioeconomic disparities. We show that disparities are largely driven by disparate use of the online holds system, which allows library patrons to receive books from any other branch through an online portal. This system thus leads to a large outflow of popular books from branches in lower income neighborhoods to those in high income ones. Finally, we illustrate the use of our model and insights to quantify the impact of potential interventions, such as changing how books are internally routed between branches to fulfill hold requests.