You're close - It was just before the 1920s that underarm hair removal once again became the norm. Both women and men have been removing unwanted body hair since the era of the caveman, with the relevant areas (head, legs, eyebrows, etc.) changing according to fashion.
For our purposes, 1915 was the year the first women's razor came out. It was also the year Harper's Bazaar ran an ad for X Bazin Depilatory Powder featuring a model with hairless underarms:
I'm sure they sent out may generous samples that year! The ritual of shaving underarms, at least in North America, has continued ever since.