They have called for an extraordinary general meeting of the National Conservative Convention to discuss the Prime Minister’s leadership of the party.
A non-binding vote is expected to be held at the meeting, which would – if it showed a lack of confidence – put pressure on the 1922 Committee of Tory MPs to find a way of forcibly removing the PM from office.
Mrs May has come under heavy criticism for her handling of the Brexit process, but survived a vote of confidence from her MPs in December.
Under party rules a year has to pass before another vote can be called, but an extraordinary general meeting must be held if more than 65 local associations demand one via a petition.
Dinah Glover, chairwoman of London East Area Conservatives, who has organised the petition, said: “I am extremely sad that we have had to organise this petition to ask Mrs May to consider her position, but it has become increasingly obvious over the last year that she has become the block to Brexit rather the solution.
“Chairmen have signed the petition because they fear Brexit will not be delivered under her leadership.
“We need a Prime Minister who believes in the benefits that Brexit can bring to our country to lead us in the negotiations and out of the EU.
“That is the only solution which can satisfy our party and the British people. Enough is enough.”
While not legally binding, a no-confidence vote at the Convention would heap political pressure on Mrs May to quit her post within weeks.
If Mrs May chose to resign or was forced out, it would trigger a Tory leadership contest with MPs selecting two of their number for a vote by party members.
The former foreign secretary Boris Johnson is the favourite to replace Mrs May as leader among the party’s grassroots, according to a new poll.
Almost a third of party members – 32.4 per cent – backed the pro-Brexit Tory to take over the helm of the party, up by 10 points in the past month.
Ex-Brexit secretary Dominic Raab was in second place with 14.7 per cent support, according to the poll of 1,128 panel members by the Conservative Home website.
Environment secretary Michael Gove, who also supported the Leave campaign, came third, ahead of foreign secretary Jeremy Hunt and home secretary Sajid Javid.
It comes after a devastating Survation poll of 781 Tory councillors found the party is in meltdown over Brexit.
Only 52 per cent of the Conservative councillors plan to vote for their own party in the EU elections, with 40 per cent planning to back Nigel Farage’s Brexit Party.
Some 96 per cent of councillors said the Brexit deadlock damaged the party.
And 76 per cent wanted the Prime Minister to resign, with 43 per cent saying she must go immediately.
The open revolt will make sobering reading for the Prime Minister as the Tories defend more than 4,000 of the 8,000 seats in local elections on 2 May.