This reminds me of a question that was asked some time ago, and was one of my favorite I've seen on this sub. We never answered it perfectly, but there was a Google answer that did a decent job.

Luckily, it seems your question has a clearer answer!

You are indeed right, that it appears to have originated during the Vaudeville era, right before the rise of cinema. In fact William Hammerstein (yes, Oscar Hammerstein's father), a very successful Vaudeville producer is sometimes credited with inventing it. However, like any good joke, the credit is widespread (and the sources for any inventor is suspect at best). Canadian medicine show performer Thomas "Doc" Kelley is also widely credited with inventing the gag, having seen an angry cook throw a partially-eaten pie at a kitchen worker.

What is clearer though, is the first instance of the joke on film: it takes place in a short slapstick film from 1909, Mr. Flip, where Ben Turpin gets pied right at about 03:37.

Within a few years, the joke had taken off, to the point that it is now one of our most recognized trademarks of slapstick humor!

And, side note, just because I love Chaplin, he was responsible for the first proper pie fight, at 18:45 in Behind the Screen (1916). You can see how quickly the gag caught on!