The New Yorker Cartoon Caption Contest is certainly the most famous and popular of the current You-Write-the-Gag exercises. So popular it has birthed a book, a game, and winners become news stories for their local newspapers.
Cartoon Collections runs the similar idea. So similar it uses New Yorker cartoonists to supply the drawings. This is a natural evolution since Bob Mankoff who runs Cartoon Collections was The New Yorker cartoon editor when the The New Yorker contest began.
These days a number of editorial cartoonists create comic art for their newspapers asking readers to write funny lines with the best of them being published.