Nominalization of verb 分く (waku, “to split, to divide”), from the 連用形 (ren'yōkei, “stem form”) of waki. Came to refer to the space between where two things split off from each other, and then, by extension, came to refer to the side of something.
the side of something (this sense may also be spelled 傍 or 側).
waki kara kuchi o dasu
“to put one's mouth in from the side” → to jump into a conversation
(Can we date this quote?), 川本三郎 (Saburō Kawamoto), “高層ビルの足元にムラがある (Kōsō Biru no Ashimoto ni Mura ga Aru, “Villages at the Feet of Skyscrapers”)”, 風の旅人 (Kaze no Tabibito, “Wind Travelers”), volume 5, Tōkyō: Eurasia Travel Co., Ltd., ISSN9784901919081, retrieved on 2013-06-17, page 103:
To the side of large buildings, streets remain that are like old-time markets, with rows of small restaurants.
脇からはそうもあるまいと見ゆるもの (わきからはそうもあるまいとみゆるもの, waki kara wa sō mo aru mai to miyuru mono): “from the sidelines, it might not look like it could be that way” → it's generally difficult for someone uninvolved in a serious matter to understand all the particulars
脇を詰める (わきをつめる, waki o tsumeru): “to fill in the underarms” → to become an adult (from the way that children's clothes would have openings in the underarms and along other seams, to allow for alteration as the child grew)
脇を塞ぐ (わきをふさぐ, waki o fusagu): “to close off the underarms” → to become an adult (from the way that children's clothes would have openings in the underarms and along other seams, to allow for alteration as the child grew)