Retrocomputing Stack Exchange is a question and answer site for vintage-computer hobbyists interested in restoring, preserving, and using the classic computer and gaming systems of yesteryear. It only takes a minute to sign up.
At the heart of the Atari 2600 was the TIA video chip, designed by Jay Miner, well known for his later work on the chips for the Atari 400/800 and the Amiga.
Who manufactured the TIA? As an in-house custom design, it couldn't be simply purchased off the shelf; I haven't seen any mention of Atari owning a chip fab, and hiring such wasn't as routine then as it became in later decades. Did they make a special arrangement with a chip company?
There were 12 different versions of the TIA, and it was manufactured by a whole host of different manufacturers, some of which are well-known. The manufactures included AMI, Synertek, Motorola, and National Semiconductor.
This article has lots of details, including pictures of the many versions of the TIA IC with many different manufacturers' stamps.