The row follows the suspension of former leader Jeremy Corbyn and suspicion among some over the direction in which Sir Keir wants to lead Labour.
They want an online conference to be held alongside Labour’s women’s conference in June.
The decision ultimately lies with Labour’s ruling national executive committee.
Mr Burgon, the MP for Leeds East, told HuffPost UK that under Sir Keir’s leadership divisions had widened while the left of the party had suffered a “wave of unjust suspensions” of party members.
He added: “Our party should be laying out a clear alternative to the Tories. But instead of uniting to fight the Tories, the leadership has been fighting the Left with a wave of unjust suspensions and a clampdown on party democracy.
“It’s clear we can’t go on as we are. Members are the beating heart of our party–volunteers who give up their spare time to keep the party going in our communities. But members are leaving in droves and many more are demoralised with the direction of the party.”
He added: “A recall conference is needed to put an end to the wave of suspensions on party democracy and to unite the whole party behind the policies we need to force the government to change track from its disastrous handling of this crisis.”
Mr Corbyn was suspended by Labour following comments he made in the wake of a report by the Equality and Human Rights Commission into antisemitism within the party.
The former leader was later readmitted into the party, but not given back the party whip in parliament.
Sir Keir faces a key electoral test in May, in a series of election around the UK.
The Conservatives hope a so-called ‘vaccine bounce’ will give their party a boost at the polls.
An IpsosMori poll for the London Evening Standard found the Conservatives have pulled ahead of Labour by four points. But Sir Keir scored better on leadership qualities than his predecessors Mr Corbyn and Ed Miliband.
In response to the poll, Sir Keir acknowledged his party had a “mountain to climb” after its crushing defeat in the 2019 general election.
But he insisted that Labour’s “priorities are in the right place”.
On a visit to Basildon town centre, Sir Keir said: “We’re now getting to a position where on the polls we’re about even, so that’s a step in the right direction.
“But we’ve got a long way to go between now and 2024 and we’re going to be working hard at this with real determination, every day, every week, every month, every year, into that election in 2024.”
On concerns over his leadership, he insisted that the “vast majority” of his party was behind him.
Last week Sir Keir was forced to apologise to Boris Johnson after he falsely denied having previously supported the UK staying a member of the European Medicines Agency after Brexit.
Sir Keir said he had “misheard”.
At the weekend his shadow attorney general Lord Falconer also had to apologise for describing the pandemic as “a gift that keeps on giving” for lawyers.