In 2007 the government diverted the funds to help cover the costs of constructing the Olympic Park, promising to repay it from the sale of the assets - a promise which so far has not been kept.
The stadium built with the funds is currently hosting the World Athletics Championships, which began on 4 August and will run until 19 August.
Jay Kennedy, director of policy and research at DSC, said: "Ten years without a Big Lottery Refund is a shameful milestone. How can we celebrate the ‘Olympic legacy’ and events like the WAC 2017 when this debt is still owed to communities across the UK?
"Hundreds of millions of pounds were effectively stolen from charities that depend on Big Lottery Fund grants to help vulnerable people and communities across this country, to subsidise the costs of venues on the Olympic Park."
The 2010 coalition government and the subsequent Conservative governments have reaffirmed the pledge to repay and the assets are now owned by the London Legacy Development Corporation, which is responsible for the redevelopment of the park.
The Government has said repayment will start in the early 2020s, but a statement from the DSC said tens of thousands of good causes had already been sacrificed for the Olympics because the money had been diverted from BLF and the money could be used to help them if it was repaid sooner.
The DSC is leading the Big Lottery Refund campaign which calls for an immediate return of the lottery cash and which it says is supported by more than 3,800 charities.
Kennedy said: "The new government should pay back the Big Lottery Fund immediately, and take on the role of creditor for repayment from assets sales from the LLDC.
"This could help tens of thousands of charities and millions of people during very difficult times.’
A spokeswoman for the Department for Digital, Culture, Media and Sport said: "We have always been clear how this money will be returned.
"We are committed to paying back the £425 million the Big Lottery Fund contributed to support the London Olympic and Paralympic Games, with payment due to start in the early 2020s."