The outlay to modify the Olympic venue has risen from £272m to £323m.
The investigation will look into inherited issues, including the cost of moving retractable seating.
"This is a breakthrough in our long campaign to highlight the intolerable burden on taxpayers," read a statement from the Olympic Stadium Coalition.
"We have never objected to the idea of West Ham playing at the stadium; the question is simply one of how much the club should pay, and how much the taxpayer should be expected to fund."
West Ham, having won the right to be anchor tenants for the Olympic Stadium in 2013, played their first game there in August, after 112 years at Upton Park.
The Premier League club contributes £2.5m a year in rent, after paying a one-off £15m towards the conversion cost.
The Olympic Stadium Coalition, which is comprised of 14 supporters' trusts and groups, said taxpayers deserved to be made "fully aware" of how much they are paying.
It added: "We hope to be able to meet the mayor to present him with the significant information that we have gathered about the operational costs of the stadium.
"The mayor's investigation is good new starting point, but it is only a start."
The investigation is "fully supported" by Mayor of Newham Sir Robin Wales, whose borough co-owns the stadium after investing £40m.
And John O'Connell, chief executive of the Taxpayers' Alliance, said: "For too long the details of this shabby deal were kept in secret and lacked proper scrutiny, so Sadiq Khan is absolutely right to look again at the case.
"It's now down to those in charge of the inquiry to get to the bottom of how West Ham were gifted the deal of the century while ensuring their investigation doesn't end up costing the taxpayer as much as the stadium."
West Ham's move to the London Stadium has been marred by fan violence, leading the club to issue a five-point security plan to prevent further disorder.
Campaigners have welcomed the start of an investigation by London Mayor Sadiq Khan into a £50m rise in the cost of converting West Ham's London Stadium.