The phrase seems to be of Scottish origin. As Hugo found, most of the earliest uses of the phrase have to do with alcohol consumption. I did find this earlier figurative use of the phrase however from The History Of The Church And State Of ScotlandThe History Of The Church And State Of Scotland, 1753 (date checkdate check):
It appears the figurative sense of under one's belt to mean owned or "contained by" goes back even further as evidenced by this old Scottish saying from A Complete Collection of Scotish ProverbsA Complete Collection of Scotish Proverbs, 1721: