The former Work and Pensions Secretary said she has secured enough support to launch a leadership campaign.
Her successor, Amber Rudd, dodged questions over whether she would put herself forward and former chief whip Mark Harper also refused to rule out a tilt at the top job.
Ex-Cabinet minister Priti Patel fuelled speculation about her ambitions by giving a wide-ranging speech setting out a vision for the future of the Conservative party.
Mrs May has yet to set a date for her departure but Ms McVey is the third Tory openly to declare their ambition to be Prime Minister, with Andrea Leadsom and Rory Stewart already putting down markers.
But more than a dozen senior figures in the party are also laying the groundwork for a leadership campaign.
Former foreign secretary Boris Johnson remains the bookmakers’ favourite. Cabinet ministers Michael Gove, Jeremy Hunt, Sajid Javid, Matt Hancock and Penny Mordaunt are all potential contenders.
Former Cabinet ministers Dominic Raab and Justine Greening are also likely to vye for the top job.
Ms McVey said she had made the decision to run after sounding out colleagues to see if they would back her.
She said: “People have come forward and I have got that support, so I will be going forward.
The former television presenter said the Conservative Party needed a leader who “believes in Brexit”, and had “belief in the opportunities” it could bring.
Ms McVey told TalkRadio that Mrs May’s departure should be handled in a “dignified and graceful” way.
“We all know Theresa May is dutiful,” she said. “She has worked for public service for many years.”
Ms Rudd set out her vision for a compassionate Conservative party with a speech announcing plans to cut the maximum financial sanctions for benefit claimants who fail to meet the conditions of their payments from three years to six months.
Ducking questions about her future ambitions, the Cabinet minister said Mrs May should be given time to meet her commitment to complete the first stage of Brexit before standing down.
She added: “We need to hold our nerve and allow her to do that. Brexit is a complicated procedure and the numbers in the House of Commons make that even harder.
Ms Patel, a former International Development Secretary, said the next generation of political leaders to “take back control”.
“That means a new generation of political leaders who trust and believes in the people and the enterprises of this great nation,” she said.
Mr Harper, a former chief whip, sidestepped questions about his ambitions in an interview with The House.
“Our focus should be on leaving and the question about who comes after her should be a question for then as opposed to a question for now,” he said.
Justice Secretary David Gauke said it would be a “mistake” to change the leader now and ruled himself out of a future contest.
The Remainer also took a thinly-veiled swipe at Mr Johnson, warning the Tories “to avoid the temptation” of becoming right-wing populists.
In a speech about the future of the party to the Conservative think tank Onward, he said: “We have to avoid the temptation to be a populist party. That would narrow our support, narrow our base and result in policies that I don’t think would be good for the United Kingdom.”
Mr Gauke also blamed Brexit supporters for fuelling the frustration of voters by promoting “wishful thinking” and ignoring the difficulties of leaving the European Union.
“Reluctance by some participants in this debate to accept that some choices have costs has meant that the debate on our future relationship has been, too often, characterised by wishful thinking.”