The Oxford English Dictionary traces the evolution of the phrase 'letting off steam' from 1768 to 1979. It started off in 1768 as referring to the actual act of letting off steam from an engine. The first reference that uses 'steam' in an emotional sense seems to come from Frederick Marryat · Snarleyyow, or The dog fiend' in 1837: "The widow..sat..fuming and blowing off her steam."

The OED doesn't necessary always have the first ever use of a word, but this seems to indicate that the phrase evolved from a technical term to an emotional/figurative one during the mid-industrial revolution.