Cubby Broccoli launched the iconic franchise with co-producer Harry Saltzman, and was joined by Wilson as a producing partner for 1985's A View to a Kill.
Barbara took over from her father to join Wilson as a producer for 1995's GoldenEye, and the pair have overseen every Bond film since through their Eon production house.
They kept tight control of the character - something that was acknowledged by Craig when he presented them with honorary Oscars in November. "Over the years many people and organisations have tried to put their own footprint on Bond," he said.
"Barbara, Michael, I can't tell you how much I admire your integrity in holding on to your singular vision as you brought Bond into the 21st Century with a passionate and a protective determination to honour the heart of this franchise."
In 2021, when Amazon agreed to take over MGM, which shared the rights to Bond with Eon, Broccoli and Wilson said they would keep creative control.
However, they have now relinquished the reins, and the deal raises the prospect of a rethink and an expansion of the franchise.
The US giant could commission more streaming spin-offs, for example, such as how the Star Wars universe has been expanded on Disney+.
Mike Hopkins, head of Prime Video and Amazon MGM Studios, said: "We are honoured to continue this treasured heritage, and look forward to ushering in the next phase of the legendary 007 for audiences around the world."
Last year, Amazon launched the first spin-off TV series, 007: Road to a Million, hosted by Succession star Brian Cox.
It featured nine teams in a race to win £1m each through a series of spy-themed challenges. However, that series was poorly received by some fans and critics and Cox later joked that he signed on to the project thinking it was the next James Bond film.
News of Thursday's deal helps to explain why there has been a long delay without any announcement about a 26th official Bond film, or its star.
Given how much time it takes to make a blockbuster movie, the gap between No Time To Die and the next film could break the six-year record for the longest period between Bond releases.
Last August, the Telegraph's chief film critic Robbie Collin wrote that there was "no script, no title, not even a director" for a new instalment.
In October, Amazon MGM Studios boss Jennifer Salke told the Guardian there were "a lot of different ways we can go" with the franshise, and that "we don't want too much time between films, but we are not concerned at this point".
She also said the company had "a good and close relationship" with Broccoli and Wilson.
However in December, the Wall Street Journal reported that "the relationship between the family that oversees the franchise and the e-commerce giant has all but collapsed".
The newspaper also claimed that Broccoli had privately described Amazon as "idiots".