Forest Green, who were relegated from League One last season, said earlier on Tuesday that Ferguson had left his role after six months in charge.
Dingley's first game in charge will be a friendly at non-league side Melksham Town later on Wednesday.
"I'm really excited for this next step of my career," Dingley, who has a UEFA pro licence, told the club's website. "I am grateful for the opportunity to step up and to lead such a progressive and forward-thinking club."
Dingley joined the Gloucestershire club in 2019 to take charge of the academy and remains the only woman to manage a club academy in the English men's league soccer. She also launched a girls' academy in 2021.
She previously coached women's teams at Nottingham Forest, Leicester City and Lincoln and was head of coaching at League One side Burton Albion's academy.
"Hannah was the natural choice," club chairman Dale Vince said. "She's done a fantastic job leading our academy and is well aligned with the values of the club.
"It's perhaps telling for the men's game that in making this appointment on merit, we'll break new ground."