At a joint press conference with MPs supporting a referendum on the terms of Brexit, Mr Blackford said that “if Jeremy can’t… then we as the leaders of the other opposition parties must rise to that challenge”.
Opposition parties said a vote of no confidence would pave the way to a so-called People’s Vote, but the Labour Party has said a motion to topple Mrs May will only be lodged when it is certain to pass the Commons.
The row over tactics appears to be intensifying, with Labour pulling out of a planned meeting between Mr Corbyn and Mr Blackford over what it sees as SNP gameplaying, The Scotsman understands.
Conservative MP Anna Soubry challenged Mr Corbyn to bring a motion of no confidence in her own government, saying: "The biggest obstacle to a People's Vote at the moment is Jeremy Corbyn.
"If not now, when, Jeremy? He has got to start this process now."
The Conservative backbencher, who appeared alongside Mr Blackford and Liberal Democrat leader Sir Vince Cable, Green Party MP Caroline Lucas, Labour MP Dame Margaret Beckett, and Plaid Cymru Westminster leader Liz Saville-Roberts, later clarified that she would not vote for a no confidence motion.
“With only weeks left before the UK is due to leave the European Union, the Brexit process at Westminster is in gridlock and the UK is in the middle of a constitutional crisis which is unparalleled in modern history,” Mr Blackford said.
“Our message to Jeremy Corbyn is that all of us have to work together to move that motion of no confidence.
“I appeal to Jeremy to do that, but all of us working together have got to accept our responsibilities, and if Jeremy can’t put himself in that position, then we as the leaders of the other opposition parties must rise to that challenge, and we must put down that motion of no confidence.”
Asked how much time Labour had to clarify its position, Mr Blackford said: "I think Jeremy has until the close of business today.”
Meanwhile, the government has confirmed that a vote on the Prime Minister’s Brexit deal will be held “before 21 January” - under legislation, the final date for a Commons debate on the UK’s terms for leaving the EU, whether a deal has been negotiated or not.
The Prime Minister's official spokesman said Mrs May would observe the "spirit" of the EU Withdrawal Act, which requires the Prime Minister to make a statement to the Commons "before the end of 21 January" if no agreement in principle has been reached with Brussels.
There was confusion at Westminster on Monday over whether the 21 January deadline applied, as a withdrawal deal has been reached.
However, responding to an urgent question from Labour MP Yvette Cooper, the Brexit department minister Robin Walker also said it would hold a vote before 21 January.
Speaking at the People’s Vote press conference before the assurance was issued, Mr Blackford said: “We know exactly what the Prime Minister is up to, because we’re now aware that we may not be looking at a vote until close to, if not on the date of 21 January next year.
“We should be very aware of the threats that come from that. What the Prime Minister is seeking to do is make this a binary choice between her deal and no deal.
“We should be saying unequivocally: that is quite unacceptable.”