Walt Disney Co. said it signed a deal with Sony Corp.’s PlayStation Vue streaming video service to carry all of the media powerhouse’s channels, including ESPN and ABC, and smaller channels like Fusion and Longhorn Network.

The deal includes an “extensive” on-demand library from Disney and will, for the first time, offer the company’s channels over the Internet on a “multi-stream” basis—meaning that more than one person in a household can stream the programming at once. Disney’s first Web TV distribution deal, struck with...