Well removing body hair has come in and out of fashion for millennia. So we can talk about how New Kingdom Egyptians (and other certain peoples) removed their underarm hair; but I'm guessing you are more concerned about more modern practices.

Modern armpit shaving in the West seems to have been launched by changing women's fashions. In the mid 1910s sleeveless dresses became fashionable for the first time in centuries. Around the same time Gillette launched a line of razors marketed towards women. And in 1915 Harper's Bazaar published an article about how women who who wore the new fashion should remove their underarm hair to maintain a clean look. So new fashions demanded it at the exact same time as it became relatively cheap and easy to do.

By the 1920s the new trend seems to have caught on. And while there have been counter fashions at some times in some western nations, underarm hair removal has been the standard in much of the West since the 1920s

A further question I'd be interested to know the answer to. Boris Johnson, in his book on Rome, says roman men used tweezers to take off all their body hair and thought anyone that didn't was uncivilised. Is this true, and have there been other periods when men's body hair was a social no-no?

It's not the norm everywhere, so are you talking about the U.S.?