Shah joined the club in 2013 and played a role in Lee Power gaining full control of the club.
In a statement, Chairman Power said: “I would like to sincerely thank Sangita for all of her efforts during her time with us.
“You couldn’t put a price on all her hard work and help during the club’s ownership battle and her knowledge was invaluable when it came to turning the business from one that lost money into one that is now breaking even.
“On behalf of everyone at the football club, I would like to thank Sangita once again and we wish her well for the future.”
A reason for her departure has not been disclosed.
Clem Morfuni has been appointed as the club's non-executive vice chairman.
Morfuni is the Founder of Axis Group, who have recently renewed their shirt sponsorship deal with Swindon
Speaking to the club website, he said: "It will be great for both parties and being in the building services industry around the world, because we’ve got offices in London, New York, New Zealand, Australia and Thailand, we saw that there is a lot of potential here with the development of the stadium.
“I spoke to Lee Power and I said that we can help the club and we can go through the processes and management of building a world class facility for the fans and the community of Swindon.
“We want to build this club back to where it should be.
“There’s a few ideas that I’ve got so I’ll sit down with Lee, discuss what he wants to do and then follow the process and start getting the ball rolling.
“The main focus for me is to look at the development and we’ve got over 700 employees worldwide, so I’ve got a whole infrastructure that can help us.
“It is all about infrastructure and you’ve got to have the right people and the right support for everyone for us to be where we want to be.
“I believe that we should be way above where we are but things happen and it is now all about rebuilding, refocusing and taking small steps in the right direction.”
Axis have an impressive portfolio having worked on Wembley Stadium, while they are currently involved in the construction of Tottenham Hotspur’s new state-of-the-art home, in addition to a vast array of other projects around the world.
“We’ve worked at Wembley and we are also doing Perth Stadium in Australia,” he added.
“We’ve built stadiums in Sydney, we build hotels, office blocks, hospitals, we’re doing stuff in New York so we are in a lot of places.
“I love football, I love construction and I think it is a good match for Lee and I, and hopefully we can build a world class facility here for the community and for the Swindon fans.
“We want to be here for another 100 odd years and that’s what we’re trying to do.
“There is so much potential here at Swindon.”
Speaking about the appointment, ower added: “We are delighted to welcome Clem on board as Non-Executive Vice Chairman.
“During my time at the club we haven’t had chance to have a detailed look at developing The County Ground, but Clem’s arrival will help with that and we are thrilled to have someone with his experience and knowledge join us.
“Clem will move discussions forward with the council and other parties and he will also be looking to explore every option when it comes to redeveloping The County Ground.”