Amazon.com Inc. and Walt Disney Co. are at loggerheads over terms for carrying the entertainment giant’s apps in Amazon’s Fire TV devices, a dispute that highlights the new power struggles emerging in the streaming economy.

Amazon is pushing for the right to sell a substantial percentage of the ad space on Disney apps, and Disney has so far resisted, according to people familiar with the situation. Some of these people say they are optimistic the companies will reach an agreement. If they don’t, Disney apps could be removed...