In the sense we think about it nowadays, in the 1960s. Second-hand textile shopping obviously has a much longer history.

A good book on this topic is eds. H. Clark and A. Palmer, Old Clothes, New Looks.

Straight-up wearing clothing from 20+ years ago to make a fashion statement was not really done until the late 1970s-1980s. Prior to that, wearing old clothes for the most part was simply wearing old clothes - but in the late 70s there was an explosion of interest in vintage fashion. While, as /u/colevintage points out in the answer to this question points out that fashion has been cyclical for a long time, it's at this point that the twentieth century really started to crib from itself. The Laura Ashley romantic/Victorian/pioneer style was fashionable at this time, and it lent itself to cannibalizing petticoats, slips, and chemises from the 1900s and 1910s or ruffled and long-skirted early 20th century dresses. Then in the 1980s, the 1940s were a strong influence on mainstream fashion, leading people to look to that decade for actual vintage pieces as well.