Margaret Hodge has described the Charity Commission as not "fit for purpose" after a highly critical report warned that its failure to investigate fraud and abuse was undermining public faith in good causes.

The National Audit Office said the commission was in need of "radical change" and accused the organisation of being too passive and slow to react.

Commenting on the report, Hodge, the chair of the public accounts committee, said the commission "risks undermining public trust" in charities. Two other members of the committee have said privately that it should be either dismantled or completely restructured. The Cabinet Office has launched a consultation so that gaps in the commission's authority can be plugged in line with the NAO report.

The organisation has faced years of criticism for failing to clamp down on abuses and commissioners have been warned to expect "tough questions" when they appear before the Commons public accounts committee later this month.

The inquiry was prompted by MPs' concerns over a complex tax avoidance scheme operated by the Cup Trust, a registered charity. The charity gave £152,292 to good causes while attempting to claim £46m back from the tax authorities in Gift Aid on £177m income. In 2010 it attracted more donations than the Royal Society for the Protection of Birds, the British Heart Foundation or the Salvation Army.

George Osborne introduced a cap of £50,000 on charitable tax relief last year after identifying widespread abuse. But an outcry from philanthropists persuaded the government to change its mind.

The commission launched a full investigation in April, admitting that the outcry over the case had been "a disaster for the charity sector" and has identified 13 other charities requiring investigation.

The NAO backed complaints that the commission should have acted against the Cup Trust sooner – suggesting the case was symptomatic of greater failings within the commission. The NAO report concluded the commission was "not regulating charities effectively" and was failing to provide value for money to the taxpayer for its £22.7m budget.

"It does important and necessary work and its independent status is highly valued, but it does not do enough to identify and tackle abuse of charitable status," it said – noting that the NAO had "repeatedly identified concerns" in reports stretching back more than 25 years.

"It uses its information poorly to assess risk and often relies solely on trustees' assurances. Where it does identify concerns in charities, it makes little use of its powers and fails to take tough action in some of the most serious cases," it added.

In the three years to March, not one charity trustee had been removed by the commission, only two were suspended and there were just 17 instances where it intervened to prevent charities entering transactions, the report pointed out.

Recent efforts to deal with a 40% cut in its budget have also been criticised by the report. The commission has failed to "put sufficient emphasis on the rigour of its registration process and its investigation of wrongdoing in charities," the NAO said.

In an accompanying report into the Cup Trust case, the NAO said the commission failed to check the organisation met the legal requirements when it registered as a charity in April 2009. Its sole corporate trustee was a company called Mountstar, registered in the British Virgin Islands.

In 2011, despite concerns raised by HMRC the commission failed to open a statutory investigation and remained "too passive", the NAO said.

"The commission took too narrow a view of its remit... It did not fully appreciate the scale and nature of the Gift Aid scheme nor the potentially detrimental impact of the case on public confidence in charities," the report concluded.

The NAO report said the commission's internal investigations had identified "potential regulatory issues in 13 charities and is looking into these issues, but has concluded that it is unlikely that there are charities similar to The Cup Trust on the register," it said.

Hodge said: "Our inquiry into the Cup Trust raised serious questions about whether or not the Charity Commission is fit for purpose. This report suggests it is not. People in this country are hugely generous in giving to charities but the failure of the Charity Commission to detect and tackle abuse effectively risks undermining public trust in the whole sector."

A spokesman for the Cabinet Office said: "As the NAO notes, the Commission should exercise the power it has more often. However, in line with NAO recommendations, the Cabinet Office will also launch a consultation to address any gaps in its current legal authority."

Responding to the report, Sam Younger, the commission's chief executive, who has been in the post for three years, accepted the findings and claimed that progress has already been made. "The commission is committed to an urgent programme of reform. The commission has already taken steps to improve our approach to tackling the most serious cases of abuse and mismanagement in charities," he said.