The claims centre around Vote Leave’s links to another campaign - BeLeave - which it helped fund.
Shahmir Sanni, who worked on the campaign, claimed Vote Leave used it to get around strict spending limits set by the Electoral Commission.
Vote Leave has strongly denied wrongdoing and said the £625,000 donated to BeLeave was within the rules.
But Mr Sanni told Channel 4 News: “I know that Vote Leave cheated... I know that people have been lied to and that the referendum wasn’t legitimate.”
Cabinet ministers Boris Johnson, Michael Gove and Chris Grayling were among senior politicians involved in the Vote Leave campaign.
Deputy Labour leader Tom Watson said: “The leaders of Vote Leave are senior cabinet ministers. We need answers about what really went on.”
But Mr Johnson said Vote Leave won “fair and square - and legally”.
Mr Sanni was in a relationship with senior Vote Leave figure Stephen Parkinson - now Theresa May’s political secretary - at the time of the referendum and claimed through his lawyers he was “outed” by the Number 10 aide in the run-up to the Channel 4 disclosure.
Mr Parkinson said he was “saddened” by the “factually incorrect and misleading” statements by Mr Sanni and his lawyers.
Liberal Democrat Brexit spokesman Tom Brake said police should be called in.
“These allegations are stunning and touch directly on one of Theresa May’s closest advisers,” he said, adding: “These allegations must be examined by the police.
“If they represent what happened it is outrageous and shameful.
“The referendum had a very narrow outcome. One of the biggest exercises in democracy must not turn out to be one of Britain’s biggest electoral frauds.”
The allegations centre on Mr Sanni’s claims that BeLeave was controlled by Vote Leave rather than an independent campaign.
Mr Sanni said: “In effect they used BeLeave to over-spend, and not just by a small amount... Almost two-thirds of a million pounds makes all the difference and it wasn’t legal.”
But Mr Parkinson said: “At the relevant time during the referendum period, the commission advised Vote Leave that it was permissible to make a donation in the way it proposed to do to BeLeave.
“Twice since the referendum the commission has investigated this matter, and twice it has found no evidence of wrongdoing. A third investigation into the same issue is currently taking place.
“The Electoral Commission has not contacted me in relation to any of these inquiries, but I will of course be happy to assist in them if they wish me to do so.
“I firmly deny the allegations in the programme.
“I had no responsibility for digital campaigning or donations on the Vote Leave campaign, and am confident that I stayed within the law and strict spending rules at all times.”
Mr Grimes also denies all the allegations, Channel 4 said.
A Vote Leave spokesman said it had “twice been cleared on this matter by the Electoral Commission” adding: “As has been the case throughout, Vote Leave is obligated to review - to the extent it can after this long elapsed period since the referendum - all such allegations, and is doing so.
“We will as appropriate share any relevant findings with the Electoral Commission, again as we have always done.”
An Electoral Commission spokeswoman said: “The commission has a number of investigations open in relation to campaigners at the EU Referendum; it does not comment on live investigations.”