Woodman has been head of goalkeeping at Arsenal since 2018, but leaves the post with immediate effect.
He will take charge of Bromley for their match against Wealdstone on Friday as he bids to help the side gatecrash the National League play-offs.
Woodman has a raft of experience as a coach, having worked at the likes of Charlton Athletic, Crystal Palace and Newcastle United.
He has not worked as a manager before, though, and has dropped down to the fifth tier of English football to cut his teeth.
“I am delighted to be given the opportunity to lead such an ambitious club and am looking forward to working towards fulfilling that ambition,” said Woodman.
“The Chairman and his vision for the club, along with the existing and planned facilities make for an incredibly exciting time ahead for players, staff and supporters.”