The 64-year-old German has led the team for the past two years.
Technical director FX Demaison will also leave his post.
Williams, owned by US investment firm Dorilton Capital, will confirm its new team principal and technical director "in due course".
"It has been a huge privilege to lead Williams Racing for the last two seasons and to lay the foundations for the turnaround of this great team," Capito said in a statement.
"I look forward to watching the team as it continues on its path to future success."
Matthew Savage, chairman of Dorilton Capital, said: "We would like to thank Jost for his hard work and dedication as we embarked on a major transformation process to begin the journey of reviving Williams Racing.
"We're grateful that Jost postponed his planned retirement to take on this challenge and now he will pass the reins on for the next part of this staged process.
"We would also like to thank FX for his contribution and wish him all the best for his future as he moves on."