In his letter to Sir Graham Brady the chairman of the Tory backbench 1922 Committee, Mr Duddridge - who served as a whip and junior minister under David Cameron - said the Conservatives need "a strong leader, someone who believes in Brexit and someone to deliver what the electorate voted for".
He said: "I write this with heavy heart, however we now need a proper leadership election and to move on".
Her main Tory rival, former foreign secretary Boris Johnson, has called the plan deranged and a sellout of Brexit.
"Let me be very clear about that, because I think there have been some people speaking about "oh well with customs everything will be across the United Kingdom, but with regulatory alignment terms we can do something there", she told the "Daily Telegraph".
May argues her so-called Chequers proposal is the only way to protect jobs and cross-border trade while also avoiding physical checks on Britain's land border with Ireland.
The government has already promised to protect the rights of more than three million European Union citizens now living in Britain after Brexit, even if there is no divorce deal.
Johnson himself held back from attacking May in person, while other critics of her European Union plan, notably Eurosceptic MP Jacob Rees-Mogg, have offered their support for her leadership.
On the final day of her party's conference, May rallied members, trying to address their concerns that the Conservatives are becoming increasingly directionless under the weight of Brexit by calling on them to look to a brighter future.
Michel Barnier has said customs controls on goods going between Great Britain and Northern Ireland could be kept light touch, through scanning technology and pre-clearance. Foster to "confront" her party's stance on equal marriage.
Theresa May suffered a blow less than an hour before her crucial speech to Conservative activists as one of her own MPs called for her to be removed as party leader.
The Manchester conference was seen as an opportunity for the embattled PM to make a fresh start, reassert her political authority and provide a vision for Brexit Britain.
■ On Labour's party conference: 'I listened carefully to Corbyn last week, and it was astonishing that he had absolutely nothing to say about the wealth creating sector of the economy... the grafters and the grifters'.
With no agreement with the bloc over the terms of divorce or a future trade relationship, the last day of the conference marked the beginning of what some officials predict will be a frenzied couple of weeks of diplomacy between London and Brussels as the two sides try to pin down a deal.
"Eventually, people in the EU think it would lead to the disintegration of the union".
Tánaiste Simon Coveney said Mrs May had given clear guarantees to protect all parts of the Good Friday Agreement around Brexit negotiations.
She will also urge Conservatives to honour their history as the "party of unionism" and back a plan that keeps Northern Ireland fully in the UK.