I've got a use of "call shenanigans" that dates to 1998, and I strongly suspect that, even if it isn't the earliest use, it's the source of the phrase's popularity over the last decade or so.
In "Cow Days," the thirteenth episode of the second season of South Park, the boys are at a carnival playing games which they come to suspect are rigged. Kyle calls shenanigans, which brings Officer Barbrady to investigate. However, the carny allows the boys to win the game when the officer is present, and Kyle retracts his call of shenanigans.
Later in the episode, the boys discover that the prizes they were trying to win are in fact cheap knock-offs, and Kyle formally reinstates his claim of shenanigans. The claim is upheld this time by Officer Barbrady, which gives the townspeople free rein to destroy the carnival with brooms.
Again, this may not be the very first use of the term, but you asked whether anyone had made it particularly popular, and I think this is what did it. I saw the episode when it was new, and I remember quite distinctly thinking that I'd never heard the word shenanigans used that way before—but within days of the airing of that episode, several of my friends were calling shenanigans every chance they got.
Actually the phrase predates Through the Looking Glass by at least thirty five years.
OED says
1836 T. C. Haliburton Clockmaker (1837) 1st Ser. 143 As large as life and twice as nateral.
This leads me to conclude that it was a catchphrase before Carroll used it, and perhaps before Haliburton used it.
Best Answer
The American Heritage Dictionary of Idioms dates for dear life to the mid-1800s, although that appears to be too late by a wide margin, as I find a use from an 1815 stage play with unquestionably the same meaning as today:
It seems clear that for dear life was a fairly well recognized idiom by the time this play was published. I found some even earlier uses, but they mostly seem to be less idiomatic and more literal.