Open App Markets Act This bill establishes rules related to the operation of an app store by a covered company (i.e., the owner or controller of an app store with more than 50 million U.S. users). An app is a software application or electronic service that may be run or directed by a user on a computer or mobile device. An app store is a publicly available website, software application, or other electronic service that distributes apps from third-party developers to users. The bill prohibits a covered company from (1) requiring developers to use an in-app payment system owned or controlled by the company as a condition of distribution or accessibility, (2) requiring that pricing or conditions of sale be equal to or more favorable on its app store than another app store, or (3) taking punitive action against a developer for using or offering different pricing terms or conditions of sale through another in-app payment system or on another app store. A covered company may not interfere with legitimate business communications between developers and users, use non-public business information from a third-party app to compete with the app, or unreasonably prefer or rank its own apps (or those of its business partners) over other apps. The bill provides for enforcement of its provisions by the Federal Trade Commission and the Department of Justice, as well as through suits brought by developers that are injured by reason of anything forbidden under the bill.