Seven of the UK's leading human rights groups and privacy campaigners have demanded an urgent review of the laws being used to authorise the mass collection and analysis of data by Britain's spy centre, GCHQ.

In a letter to the Commons home affairs select committee, they call on MPs to scrutinise the way ministers and intelligence agencies have interpreted the Regulation of Investigatory Powers Act 2000 (Ripa).

Documents leaked by the whistleblower Edward Snowden have shown that the government relies on vaguely worded provisions in Ripa to justify surveillance programmes such as Tempora. This permits GCHQ to harvest millions of emails, phone calls and Skype conversations from the undersea cables that carry internet traffic in and out of the country. The information is being shared with GCHQ's US counterpart, the National Security Agency.

The practice is being challenged by Big Brother Watch, Open Rights Group, Liberty, Privacy International, Index on Censorship, English PEN and Article 19. They have written to Keith Vaz, chair of the select committee, raising their concerns.

"It appears that these powers are being interpreted far more broadly than was ever intended by parliament," the letter says. "Equally, data-sharing arrangements between GCHQ, the NSA and other agencies raise questions about the extent to which oversight structures and legal safeguards are being effectively circumvented."

The letter also cites the "disturbing reports" of the way Ripa was used to snoop on Duwayne Brooks, the best friend of Stephen Lawrence, murdered in a racist attack in 1993. Police secretly recorded a meeting he had with his lawyer during a reinvestigation of Lawrence's murder.

The groups say stories published by the Guardian showing the extent of Britain's surveillance programmes appear to undermine the need for new laws to give spy agencies more powers to access data from internet service providers.

The home secretary, Theresa May, has been vigorously championing the communications data bill, insisting there is a growing "capability gap" that is hindering GCHQ, MI5 and MI6 in their work on counter-terrorism.

The groups argue that the bill appears to have been intended to "provide a statutory basis for activity that is already under way without parliamentary approval or oversight".

"Given current international concern about surveillance, questions about parliamentary oversight and approval of GCHQ's operations and ongoing concerns about Ripa's fitness for purpose, we urge your committee to undertake wide-ranging, post-legislative scrutiny of Ripa and the wider landscape of surveillance law in Britain," the letter says.

Isabella Sankey, director of policy for Liberty, told the Guardian that the matters were too serious to be considered only by the parliamentary intelligence and security committee (ISC), the body that is supposed to oversee the work of the agencies. Critics say the ISC has limited resources and point out that it takes evidence only in secret.

"We suspect GCHQ of intercepting billions of private emails and messages belonging to the UK population without parliamentary knowledge or approval. Matters so grave for public trust, human rights and democracy itself cannot adequately be investigated by a secretive committee, taking spin from spooks in the dark," she said.

The letter to the committee is the latest attempt by campaigners to force an open debate about the work of the spy agencies in the wake of Snowden's revelations. Last week Privacy International launched a legal challenge calling for the immediate suspension of Britain's use of material from the Prism programme, which is run by the NSA.

Prism allows the NSA to intercept the communications of non-US citizens living outside America from global internet companies such as Google, Facebook and Yahoo. The group also demanded a temporary injunction against the Tempora programme. Lawyers acting for Privacy International say the programmes are not necessary or proportionate.

The claim was submitted to the investigatory powers tribunal (IPT), which is supposed to review all complaints about the conduct of Britain's spy agencies. Liberty has also complained to the IPT about the extent of the UK surveillance programmes.

The shadow home secretary, Yvette Cooper, has also called for the scrutiny role of the nine-strong ISC to be strengthened. In a speech to the Demos thinktank, she said too much of the oversight was unsatisfactory, paper-based and failed to provide assurance to parliament and the public that there would be a rigorous investigation when things went wrong.