Bradbury leaves the Sports in 16th place in the National League South after arriving at the club this summer.
It comes after Oxford City relieved Mark Jones of his duties on Monday afternoon.
Fan favourite and captain Sergio Torres will help out with first team duties, the club have said.
"The directors and football committee would like to place on record their thanks to Lee for all that he has done in his short time at the club and also for his professionalism at this difficult time for the club," a statement said on their official website.
Departing Bradbury added: "I would like to start by thanking everyone associated with Eastbourne Borough football club for the way they have treated me since day one of me joining this fantastic community club.
"I part company with a heavy heart but also know it’s best for both parties. All the best for the future. Come on you Sports!"