More than 40 cross-party MPs have demanded beefed up powers for the electoral watchdog after the official Brexit campaign was fined and reported to the police over breaches of spending laws.

The Electoral Commission imposed a £61,000 fine on Vote Leave and referred David Halsall, the campaign’s “responsible person”, to the police for making false declarations of campaign spending, after it was found to have coordinated illegally with BeLeave, another Brexit group.

In a letter to John Bercow, who chairs the speaker’s committee on the Electoral Commission, MPs said the findings show democracy could be “vulnerable to tampering and manipulation” and the watchdog needs tougher powers to restore public faith in the electoral system.

It comes amid calls from senior MPs for the Brexit referendum to be “rerun” after the sanctions against Vote Leave. Vote Leave said the electoral commission’s findings were “wholly inaccurate”.

Labour MP Stephen Kinnock, who helped to coordinate the letter, said the watchdog was an “analogue regulator in a digital age” and warned that “dark money and dark data” could flood the system without reform.

The suggested reforms include allowing the Electoral Commission to impose unlimited fines and to refer breaches to specialist police officers to investigate.

MPs also called for campaigns to be forced to declare their spending online, and outlined the need for tougher regulation for digital political advertising – something the commission itself has backed.

“Until these reforms are instituted, our democracy will remain susceptible to future abuses like these,” the letter said.

“Looking ahead, we cannot in good conscience have another election or referendum without ensuring our polls are free and fair.”

Mr Kinnock said the democratic system was under attack from a mixture of foreign influence and abuses of the laws, describing the Vote Leave result as the “thin end of the wedge”.

“The Electoral Commission is an analogue regulator in a digital age and is in desperate need of reform,” he said.

“The commission’s fines are laughable and seen simply as the cost of doing business. I hope its recent call for reform will represent a fork in the road.

“The government must act immediately otherwise abuse will continue and trust will decline – more dark money and dark data will flood the system and buy influence.”

His concerns were echoed by Green Party co-leader Caroline Lucas, who said it was “fanciful” to suggest fines of £20,000 were enough to dissuade people from breaking the rules.

She added: “I am not asking for a rerun of the referendum, but I think it strengthens the case for a people’s vote on the final Brexit deal.”

Kyle Taylor, director of the Fair Vote campaign group, which helped to organise the letter, said: “We cannot simply allow elections and referenda to be hijacked by illegal activity.

“These elections decide the UK’s destiny and we the people deserve better.

“Our politicians must now introduce measures to ensure this doesn’t happen again so we can be sure our elections are free and fair, and that the UK’s future actually reflects the true will of the people.”

The Electoral Commission welcomed support from MPs for greater powers, as the organisation says it needs the tools to regulate changes in political campaigning online.

A spokesperson said: “We have been calling for parliament to make changes for over a decade and most recently set these out in detail in our digital campaigning report.

“With the increasing use of online and targeted digital communications, action must be taken by the UK’s governments and parliaments, and by social media companies, to improve transparency for voters.

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“This includes implementing one of our key recommendations to change the law so that online materials produced by campaigners have an imprint stating who has created them.

“We also continue to call for higher sanctioning powers in order to provide an effective deterrent, and counter the risk that well-funded campaigners will come to see our current fines as the cost of doing business.”

It came after the watchdog said it found “significant evidence” of joint working between Vote Leave – which was fronted by Boris Johnson and Michael Gove – and BeLeave, a youth Brexit group run by Darren Grimes.

BeLeave received a £675,000 donation from Vote Leave, which would have taken the campaign’s spending over the £7m limit if it had been recorded as part of Vote Leave’s referendum expenditure, the commission said.