Tina Flournoy, the chief of staff for Vice President Kamala Harris, has established a reputation as a tough gatekeeper, using similar tactics to those she used while overseeing former President Bill Clinton’s post-White House office.

Flournoy’s tight approach, which includes blocking longtime allies from contacting Harris, has led to both praise and criticism.

It also has created tension within the vice president’s team and among her outside confidants and donors. Politico reported on Wednesday that there is dysfunction in Harris’ office and that much of the frustration stems from Flournoy. She oversaw...