While it’s true that William Shakespeare used the phrase “Go; vanish into air; away!” in his play Othello in 1604 and “These our actors, as I foretold you, were all spirits and are melted into air, into thin air” in The Tempest in 1610, the exact phrase “vanish into thin air” is found in The Edinburgh Advertiser of April 1822, in a piece about the then imminent conflict between Russia and Turkey:
“The latest communications make these visions vanish into thin air.”
So while there may have been others before April 1822 who used the phrase “vanish into thin air” in their works, any references I could track down have vanished into thin air, making it impossible to confirm their existence.
Historically Speaking 