Blind people frequently encounter inaccessible dynamic touchscreens in their everyday lives that are difficult, frustrating, and often impossible to use independently. Touchscreens are often the only way to control everything from coffee machines and payment terminals, to subway ticket machines and in-flight entertainment systems. Interacting with dynamic touchscreens is difficult non-visually because the visual user interfaces change, interactions often occur over multiple different screens, and it is easy to accidentally trigger interface actions while exploring the screen. To solve these problems, we introduce StateLens - a three-part reverse engineering solution that makes existing dynamic touchscreens accessible. First, StateLens reverse engineers the underlying state diagrams of existing interfaces using point-of-view videos found online or taken by users using a hybrid crowd-computer vision pipeline. Second, using the state diagrams, StateLens automatically generates conversational agents to guide blind users through specifying the tasks that the interface can perform, allowing the StateLens iOS application to provide interactive guidance and feedback so that blind users can access the interface. Finally, a set of 3D-printed accessories enable blind people to explore capacitive touchscreens without the risk of triggering accidental touches on the interface. Our technical evaluation shows that StateLens can accurately reconstruct interfaces from stationary, hand-held, and web videos; and, a user study of the complete system demonstrates that StateLens successfully enables blind users to access otherwise inaccessible dynamic touchscreens.