More than 50 MPs and peers from across the main political parties have written to the Metropolitan Police and the National Crime Agency asking them to investigate unlawful activities committed by the Vote Leave and Leave.EU campaigns.

The group demanded thorough investigations into Electoral Commission findings that pro-Brexit campaigners breached spending rules, saying it was vital to “the integrity and security of our electoral system and democracy”.

Several Conservatives signed the letter, including Sarah Wollaston, the chair of the Commons health committee, and peers Baroness Altmann and Baroness Wheatcroft.

Senior Labour and SNP MPs, and Liberal Democrats including party leader Sir Vince Cable, were also among the signatories.

They urged law enforcement agencies to investigate the Brexit campaigns’ “law-breaking”.

In their letter, the group said: “The Electoral Commission, the Information Commissioner, the House of Commons digital, culture, media and sport committee and others have been doing and continue to do important work in this area. However, their powers of investigation and the sanctions at their disposal are limited and they have no power to prosecute. This places a particular responsibility on those who do.”

The MPs and peers said the investigations carried out so far were “extremely important” but added: “So is your role and responsibility in investigating wrongdoing, enforcing our laws and protecting our democracy. This is why we are urging you to investigate these matters thoroughly and with urgency.”

Among activity they said needed further investigation was Vote Leave’s unlawful campaign spending. The organisation was fined £61,000 by the Electoral Commission for four breaches, including misreporting spending, overspending the £7m limit by more than £500,000 and filing a return that included 43 wrong items.

The overspending was partly a result of collusion between Vote Leave and another group, BeLeave, during the referendum campaign. VoteLeave was found to have channelled hundreds of thousand of pounds to BeLeave for a series of campaign adverts, helping it evade strict spending limits. Student Darren Grimes, who led the BeLeave group, has since been fined £20,000 over the matter and is facing a police investigation.

The MPs and peers also said the Leave.EU campaign, led by Nigel Farage and UKIP donor Arron Banks, should be investigated by police over numerous legal breaches, including spending “at least” 10 per cent more than permitted. The group was fined £70,000 for the offences.

The Electoral Commission has passed its findings to the police, who are considering whether criminal charges should be brought.

Ben Bradshaw MP, a former Labour cabinet minister who helped organise the letter, told The Observer: “There has been no bigger decision – beyond questions of war and peace – before the British people in the last half century. The importance of the question demanded that both the spirit and the letter of the law were respected.

“But we know that Leave did neither and have been fined by the Electoral Commission, and leading figures have been referred to the police. They ran a campaign that was so bent they probably discussed whether they could pay for their dubious and disgusting Facebook ads with £9 notes.

“Threatening the integrity of the electoral system is surely one of the highest crimes in any democracy. It is essential that the police rigorously and fully investigate these allegations. These are not minor questions of missed deadlines, wrongly ticked boxes or badly filed invoices, but matters of the gravest importance.”

Tory peer Baroness Altmann, also a former minister, added: “Democracy needs to be based on honest representation. But representations made to the people in the referendum campaign were untrue. If voters relied on the assurances and descriptions made by the Leave campaign, which were false, then democracy is not served by forcing parliament to obey that vote, regardless of the consequences once the true position is known.”