I might be wrong, but I think the most recent team to start from scratch (and is still in the sport) is Red Bull. They started out as Stewart in 1996, then became Jaguar, then Red Bull. I remember watching a really interesting documentary about Jackie Stewart getting the team up and running, and sweating about his car meeting the safety rules.
Toyota also started from scratch after this, but then left.
Most/all other teams essentially bought older teams and rebranded them.