Vinney, who joined the board in 2003, will be succeeded by Keith R. McLoughlin as chairman.
The company -- known for its iconic red and white cans of soup -- said only that Vinney "has chosen to retire from the Campbell Board due to other personal commitments."
Saying it was an "honor to serve" as a director and as chairman of Campbell, 70-year-old Vinney praised the board's vote to promote McLoughlin as chair.
"The Board is fortunate to have a strong and capable leader in Keith, whose steady direction has been indispensable over the last year," said Vinney in a statement.
Chairman since 2015, Vinney was previously chief executive of medical supply company STERIS Corp., and prior to that, a chief financial officer at Goodrich Corp.
McLoughlin, age 62, has been a Campbell's director since 2015 and served as interim president and chief executive from May 2018 to January. Mark Clouse replaced him at the start of this year.
Before joining Campbell, McLoughlin was chief executive of AB Electrolux and prior to that spent two decades E. I. duPont de Nemours and Co.
Campbell's board also brought on J.P. Bilbrey, former chief executive of The Hershey Company. Bilbrey's appointment was part of the agreement Campbell made with Third Point last year, "stipulating that the company consult with Third Point (TPRE) - Get Report on the appointment of a third director by the May 2019 Board meeting," according to a spokesperson.
In addition, the board said that member Sara Mathew, who joined in 2005, has also retired from the board.
Campbell made the announcement after market's close on Thursday. Shares ended the day up 0.92% to $38.30.
This story was updated to include additional information clarifying Bilbrey's appointment to the board.