A cross-party group of more than 50 MPs and peers has written to the Metropolitan police and National Crime Agency calling on them to fully investigate examples of “law-breaking” by Vote Leave and others who campaigned for the UK to leave the European Union.

The group, which includes Tory MPs and Conservative members of the House of Lords, says thorough investigations into “law-breaking established so far” by organisations including the Electoral Commission must be carried out “given their importance to the integrity and security of our electoral system and democracy more widely”.

The letter has been signed by Sarah Wollaston, the Tory chair of the health and social care select committee, the Conservative peers Ros Altmann and Patience Wheatcroft, the Liberal Democrat leader Sir Vince Cable, and senior Labour and SNP MPs.

It states: “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 letter then lists several of the findings of investigations conducted so far. These include:

•Arron Banks’s Leave.EU campaign being fined £70,000 by the Electoral Commission for multiple breaches of electoral law, including exceeding spending limits by “at least” 10%.

• Vote Leave, the official Leave campaign, being fined £61,000 for four breaches of the law, including misreporting campaign spending, overspending by more than £500,000, and filing a spending return which included 43 inaccurate items.

• The commission finding collusion between Vote Leave and another group, BeLeave, in the weeks running up to the referendum, which involved Vote Leave spending more than £600,000 in the name of student Darren Grimes’s BeLeave campaign.

The MPs and peers say the work of such organisations is “extremely important” but adds: “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.”

Former Labour cabinet minister Ben Bradshaw, who helped organise the letter, said: “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.”

Former pensions minister Baroness Altmann 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.”

“In our country, there are strict rules which require honest disclosure. You could not sell someone a washing machine, or a pension, on the basis of flawed representations such as those made by the Leave campaign. Those who tried to sell someone a product which was so dishonestly described would be prosecuted, and the customer would be compensated for any loss.”