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.
It’s true that this expression is in an interrogative form that is not normally found in contemporary English. It uses inversion rather than the auxiliary verb do. We would normally expect What do you have? (or, in the UK at least, What have you got?) It’s a colloquial way of saying, in the Oxford English Dictionary’s definition, ‘anything else (similar) that there may be, or that one can think of’, or even, simply, etcetera. It’s of US origin, and the Oxford English Dictionary’s earliest citation is from the 'New Yorker' magazine in 1925.
Best Answer
The OED says it's chiefly and originally Liverpool and Lancashire slang and compares it to do a bunk and do a runner. Their earliest citation is the Liverpool soap opera Brookside from 1990: