Saudi Arabia’s sovereign-wealth fund has pumped $400 million into Travis Kalanick’s new company CloudKitchens, according to people familiar with the situation, in a deal that could value the operator of so-called ghost kitchens at about $5 billion and reunites the former Uber Technologies Inc. chief with one of his biggest backers.

The Saudi fund’s agreement with CloudKitchens was completed in January, the people said. It was the fund’s first known deal in Silicon Valley since the murder of journalist Jamal Khashoggi last...