Winter, 50, succeeded Aileen Trew, the wife of former chairman Ray Trew, in October 2015.
Trew left the club last month, and announced the Magpies, currently 17th in League Two, were up for sale.
Winter, who was previously CEO at Sheffield United, appointed Jamie Fullarton as manager in January after Ricardo Moniz was sacked.
The club have stated that he will work a notice period before handing over all duties.
"There have been a lot of changes at the club over recent weeks and I want to stress that my resignation is not in light of these changes," Winter said in a statement.
"I came here with a clear plan of how to take this club forward and it is with regret that I have not been able to complete this important work."