Around 1915 sleeveless dresses became popular. This proved to be a great opening for market research folks to invent a new idea as well as a new product. In May, 1915, Harper's Bazaar, a magazine aimed at the upper crust, had an ad which featured a waist-up photograph of a young woman who appears to be dressed in a slip with a toga-like outfit covering one shoulder. Her arms were arched over her head revealing perfectly hairless armpits. The first part of the ad read, 'Summer Dress and Modern Dancing combine to make necessary the removal of objectionable hair.'
Women's razors and depilatories showed up in the Sears Roebuck catalog in 1922, the same year the company began offering dresses with sheer sleeves.
The leg hair campaign was more fitful. The volume of leg ads never reached the proportions of the underarm campaign. Women were apparently more ambivalent about calling attention to the lower half of their anatomy, perhaps out of fear that doing so would give the male of the species ideas in a way that naked underarms did not. Besides, there wasn't much practical need for shaved legs. After rising in the 1920s hemlines dropped in the 30s and many women were content to leave their leg hair alone.
Still, some advertisers as well as an increasing number of fashion and beauty writers harped on the idea that female leg hair was a curse. What may have put the issue over the top was the famous WWII pinup of Betty Grable displaying her awesome gams. Showing off one's legs became a patriotic act. That, plus shorter skirts and sheer stockings, which looked dorky with leg hair beneath, made the anti-hair pitch an easy sell.
Some argue that there's more to this than short skirts and sleeveless dresses. In fact, Greek statues of women in antiquity had no pubic hair, suggesting that hairlessness was some sort of ideal of feminine beauty embedded in Western culture.
Much of this information was culled from Cecil Adams' "The Straight Dope."