I noted that there are several versions of the word that mean different things. I'm interested in the colloquial version that is the practice of doing paid work in addition to one's regular employment (one's day job).
I searched both the OED and Webster's dictionary online and noted a slight discrepancy:
I checked the Ngram viewer and noted that their usage starts around this time but couldn't find anything that clearly pre-dates 1864.
I then began searching magazine and newspaper collections available through the UCF Libraries including the New York Times database, ProQuest's historical newspaper database, the Vogue archive, and JSTOR. While I found plenty of examples, I couldn't find one that pre-dated the OED citation.
This word is cited twice in the OED:
a1834
Friendships..The sorts, methinks, may be reduced to three. Acquaintance many, and Conquaintance few; But for Inquaintance I know only two—The friend I've mourned with, and the maid I woo!
S. T. Coleridge in Fraser's Magazine (1835) vol. XI. 54
1840
There must be a want of ‘inquaintance’ (if I may borrow Coleridge's word) with the spirit of Shakspeare's plays.
Fraser's Magazine vol. 22 613
But searches in numerous databases revealed only false positives, so it looks like this word was first used by S. T. Coleridge and then cited by another author in 1840.
I also found a reference to Coleridge from 1842 in Ainsworth's Magazine - Volume 1 - Page 339, so it seems as though that's the origin of the word and it's unlikely we'll find an older citation.
There is no Webster's citation.