He said opposition MPs had to put aside their differences and work together to stop the Prime Minister who he said, is "prepared to crash the bus" over Brexit.
Mr Blackford's remarks come a day after Mr Johnson said that the chances of securing a deal with the EU were "touch and go", and a day ahead of a Brexit cross-party meeting in Westminster organised by Labour leader Jeremy Corbyn.
Ahead of the return of UK parliamentarians to Westminster on September 3, Ian Blackford said MPs must move quickly on their return to the House of Commons in order to seize the order paper in a bid to take the option of a no-deal off the table.
Speaking on the BBC's Good Morning Scotland programme on Monday, Mr Blackford said: "We need to reflect on the fact that Boris Johnson has become Prime Minister on the votes of Conservative members of Parliament.
"He doesn't have a mandate from the voters and at no point has any party, and indeed in the referendum in 2016, was the proposition put forward that we would be leaving the European Union on a no-deal basis.
"He needs to be stopped and I'm appealing to parliamentarians this week to recognise the responsibility collectively that we've all got that when we get back to Parliament next week, we have to move fast.
"We are up against a Government that are prepared to crash the bus and they must be stopped."
Earlier this month, Labour leader Jeremy Corbyn put forward a proposal to act as a caretaker Prime Minister in the event of winning a no-confidence vote against the government. Mr Corbyn indicated he would then move to extend Article 50 to avoid crashing out of the EU with no deal in place, before calling a snap general election.
And Mr Blackford said the SNP would be ready to work with anyone in order to avert a no-deal scenario.
He said: "We will work with anybody and everybody to stop this and it's not about who becomes prime minister in the short-term - in many respects we don't even need to do that.
"What we need to do is stop no-deal and we need to stop the damage that Boris Johnson will do.
"We need to bury our differences and we need to recognise the challenge that we face from someone that's prepared to take us out of the European Union on a no-deal basis.
"We all need to work together, that's what the public would expect from us. It's about protecting the interests of all our constituents from Brexit."
Yesterday Liberal Democrat leader Jo Swinson wrote to Mr Corbyn claiming that his plan to put himself forward as a caretaker Prime Minister could jeopardise efforts to stop a no-deal Brexit because he does not have enough support in Parliament.
However, Mr Blackford suggested that Ms Swinson is "playing games".
"It's not about whether or not Jeremy Corbyn has support, it's about the responsibilities we have as members of Parliament," said Mr Blackford.
"What we need to do is bring forward legislation and I'm disappointed that Jo is playing games like this.
"All of us have to work together - it's not about the individual, it's about making sure that we put in place legislation to stop no-deal."