Although it wasn't exactly shaving. It was done with caustic gels or lathers.
If you check out the copious number of nude "art shots" and prostitute photos that have survived from the 1850-1920 era, you'll see that there was an inclination towards "Grecian purity" aka hairlessness, or....ohhhh....quite the opposite.
To judge from the surviving pornographic films of 1910- 1930 (and there are a lot of them) the class of women who were willing to ....uhhhh...give all for their art... DID engage in armpit shaving.
To judge from magazine ads of the 1890-1920 era, women were already being told, subtly and not too subtly, "You are too hairy, oh and by the way, you also smell." At least in the U.S. A horrific looking "hair stripper" was commercially available. Buried somewhere on ET is a 1926 ad I once posted that explains how you can always tell a true flapper by her elegant armpits....just as an FYI, if you are ever called upon to identify a true flapper.
BUT, at least in certain circles, men DID forbid their wives to use depilatories, since "decent women" simply did not need to shave their legs, armpits, or nether regions.
And, in other circles, women had neither the time, inclination, or cultural motivation to do so.