The letter to clients, seen by PRWeek, accuses Rudd of using office space at Finsbury to "plan and execute his boardroom coup against the staff".
The result of Rudd's actions, the authors of the letter claim, is that "one of the biggest and most effective political movements this country has ever seen has been effectively disabled during a crucial week on Brexit."
The campaign had planned to unveil a tactical voting website and send a petition to the Government and the European Parliament after the UK failed to leave the EU last Thursday.
The crisis in the People's Vote campaign – which aims to secure another referendum on the question of EU membership – became public last week.
Media reports suggested that Rudd wants to abandon the central idea of a second referendum in favour of turning People's Vote into a pro-Remain organisation, campaigning for a return to the EU if the UK eventually leaves.
Rudd was accused of taking a "wrecking ball" to the People's Vote campaign after he sent an email to staff late last month advising them that the organisation's chief executive, James McGrory, and its director of comms, Tom Baldwin, would be "leaving the organisation with immediate effect".
McGrory has been replaced by Patrick Heneghan, a former campaigns chief for the Labour party. It is not yet clear whether a replacement has been found for Baldwin.
The decision brought the campaign group to a state of civil war and staff staged a mass walkout in protest at the move.
In the letter to clients, staff claimed that, at a meeting last week, a motion of no confidence was passed in Rudd and Heneghan by a margin of 40 to three.
The letter added: "People's Vote staff just wish to get back to work but feel unable to until this situation is resolved, the serious issues raised by staff are addressed, Patrick Heneghan is removed from his position and both James McGrory and Tom Baldwin are returned to their jobs."
According to media reports, the Finsbury founder was able to fire McGrory and Baldwin after quietly taking over control of Open Britain, the largest of the organisations in the People's Vote group, using another company called Baybridge.
In a second letter, to the board of Finsbury parent company WPP, staff said Open Britain was just one of the bodies under the auspices of the People's Vote campaign.
They claimed Rudd had admitted to them that none of the other constituent organisations were consulted about the sacking of Baldwin and McGrory and neither were People's Vote campaign staff.
Rudd was asked if he accepted that the image of the People's Vote campaign had been tarnished by the sackings and whether he planned to reinstate McGrory and Baldwin.
Replying to PRWeek's enquiries on Rudd's behalf, a member of staff at Finsbury initially promised a response before later declining to comment.
However, a statement received by PRWeek from Heneghan at People's Vote read: "Last Sunday, the board of the campaign decided to change the senior executive team. They did this because they believed the campaign needed a different direction for a general election."
The statement acknowledged that campaign staff had left "in solidarity" with McGrory and Baldwin but claimed that many others had stayed on to continue its work during the general election.
The statement concluded: "Divisions like this are best resolved in private in a collegiate, calm, and professional manner. We really hope the rest of the team who aren't at work decide to come back as soon as possible and nothing prevents them from doing so."
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