The SNP’s Westminster leader’s actions come after the BBC revealed Mr Johnson personally promised Sir James Dyson he would “fix” an issue over the tax status of his employees.
Text messages seen by the BBC suggest the pair spoke after Sir James was unable to get assurances from the Treasury.
Mr Blackford said: “The revelations surrounding the Prime Minister s interference in Covid contracts are incredibly serious.
"Whether it's 'cash for questions' in the ‘90s or 'texts for contracts' during this pandemic – people know this is the same old pattern.
"The Prime Minister is at the very heart of this scandal.
"He had all the time in the world to fix contracts for a cosy club of friends and Tory donors, but didn't have any time to support the millions of self-employed.
"Those three million people didn't have a David Cameron or a James Dyson to text the Prime Minister for them.
"The Prime Minister was eager to initiate an inquiry into his predecessor David Cameron, he should be as quick to commit to a public and comprehensive inquiry into himself and his own government."
Mr Johnson appears to have now agreed agreed to publish all personal exchanges on 'Covid contracts'.
Taking place in March last year, it was at a time the UK Government was appealing to firms to supply ventilators amid fears the NHS could run out.
Sir James, whose firm is now based in Singapore, wrote to the Treasury asking for an assurance that his staff would not have to pay additional tax if they came to the UK to work on the project, and then took the matter directly to Mr Johnson.
The Prime Minister replied: “I will fix it tomo! We need you. It looks fantastic.”
When Sir James then sought a further assurance, Mr Johnson replied: “James, I am First Lord of the Treasury and you can take it that we are backing you to do what you need.”
Labour activists protested outside Downing Street on Wednesday following the revelations, which come at a time the UK Government is facing questions over former prime minister David Cameron lobbying Chancellor Rishi Sunak directly on behalf of Greensill Capital.
Angela Rayner MP, Labour’s deputy leader, said: “This government has become infected with sleaze to the point where ministers either no longer know or simply no longer care about the difference between their personal interests and the public interest.
“The public deserve answers to serious questions about billions of pounds of taxpayers’ money being handed out to Tory donors and cronies and dodgy lobbying for commercial gain.
“An urgent and thorough investigation is needed now to root out this sleaze.”
The Government said it was right to secure equipment for the NHS in “extraordinary times”.
A Government spokesman said: “At the height of the pandemic, there were genuine fears that we would quickly run out of ventilators, leaving the NHS unable to treat patients and putting many lives at risk.
“As the public would expect, we did everything we could in extraordinary times to protect our citizens and get access to the right medical equipment.”