Amazon is taking steps toward releasing a video-streaming device in time for the holiday selling season, according to people briefed on the company’s plans. The set-top box, which would pit the online retailer against a host of established rivals, is a small device that resembles a Roku Inc. player and is similarly styled as a platform to run apps and content from a variety of sources, these people said. It would also serve as a delivery vehicle for Amazon’s existing streaming video service—available as part of its Prime membership—which competes with Netflix and has been expanding lately. Plans for a device that would stream video were reported in April by The Wall Street Journal and other media outlets. Such a set-top box would broaden Amazon’s reach into the living room, where today it is dependent on other hardware makers to reach consumers watching video on TV sets.