American Buck, 76, who had been chairman since Roman Abramovich became owner in 2003, was set to continue in the role after the takeover of the club by a Boehly-led consortium last month.
A change of direction means he will now support the club as a “senior adviser”.
“Now is the right time to step down and let new ownership build on the strong foundations we have in place,” he said.
“The owners have a compelling vision for Chelsea’s future, and I look forward to helping them achieve it in this new role alongside our incredible staff, players, coaches and supporters.”
Chelsea’s £4.25bn sale to a consortium led by American investor Boehly and private equity firm Clearlake Capital was completed on 30 May.
After Russia’s invasion of Ukraine, Abramovich said in March that he planned to sell the Premier League club, shortly before being sanctioned by the UK government for his alleged links to Russian president Vladimir Putin.
The sanctions included Chelsea being placed under a strict transfer embargo and the club operated under a special licence, which allowed the sale to go ahead.
During Buck’s chairmanship, the Chelsea men’s team won 18 major honours and the women’s team won 12.
“I am proud to have helped Chelsea realise great success on the pitch and make a positive impact in the community,” he added.
Boehly said that Buck had “led Chelsea Football Club to the highest levels of international and domestic football” and thanked him for “his service and his commitment to the club”.