Vigouroux has been Orient's first-choice in the last three seasons and was a key part of their promotion as League Two champions with least goals conceded (34) in the fourth tier.
"We are looking forward to Lawrence being part of what we are building here at Burnley," manager Vincent Kompany (37) said.
"He has come through top academy systems to become the player he is today with standout performances in recent years. He is a very talented goalkeeper with a good personality and someone we will enjoy working with."
The Championship winners signed Ireland defender Dara O'Shea (24) from second-tier side West Bromwich Albion on Friday in a bid to bolster their squad for their Premier League return.
"It's amazing to be here," Vigouroux told the Burnley website. "An unbelievable training ground to be a part of and I'm really happy."
