The club suffered their second successive relegation this season, dropping out of the Football League for the first time in 106 years.
In a statement she said: "It is with considerable regret and sadness that I have decided to resign my position of vice chair lady of Stockport County."
Gibbons was informed by Greater Manchester Police
that threats had been made against her
before the final home game of the season against Cheltenham Town.
Gibbons, along with her husband Tony, have also resigned from their positions as directors at Edgeley Park.
The statement continued: "Relegation has broken all our hearts. We share the anguish it has caused, not only for the fans, but also for all those wonderful people who work so tirelessly behind the scenes.
"County's future now rests with the other board members, who we urge to act in the best interests of the club at all times.
"We are fully aware of on-going discussions that we believe will secure the club's future and provide an exciting foundation on which to rebuild County.
"We urge the current board and shareholders to do everything in their power to bring these discussions to a positive conclusion.
"Tony and I will remain County fans, and look forward to being able to watch the club we love prosper in the coming years."
Source: BBC Sport
