He replaces Steve Cunningham and will lead the Bucks in their first National League North season.
Buxton chairman David Hopkins said the new manager shares the club's ambitions: "Jamie is an ambitious young leader with experience at our new National League North level and above. We expect this to prove a great appointment."
"I could not resist the chance," said new team boss Jamie. "The ambition of the chairman and others within the club is most impressive. Coupled with the strong fan base and community involvement I was immediately drawn to the opportunity.
"The Bucks have experienced amazing league and cup success recently and now everyone is striving for more. I am looking forward to being part of that. Leaving Chorley wasn't easy but it's all about Buxton's future now. We're starting to get ready for pre-season and the great challenges that lie ahead."
A former Chorley player and in charge for four years, Vermiglio, 39, achieved one promotion, two play-offs and a club record FA Cup fourth round appearance.
He also set up a youth academy. Chorley finished sixth in the 2021/22 National League North, losing to York City in the play-offs.