I've heard this story a few times, and it seems to originate from this story. But has an appendectomy ever been a requirement for submariners of a specific nation? Or is it just a legend and an encouragement for young people ("Your belly hurts, but well, you'll be able to work in a submarine once the doctors are done!") ?
[According to the US Naval Institute, yes] an appendectomy was performed on a submarine1.
According to that site (and pretty much every google search result) the surgery was successful, and Lipes even received a special commendation for his actions.
According to NPR, a reporter even won a Pulitzer Prize for reporting on that story.
I see no reason to doubt this story.
As to requiring a person to have had one prior to serving on a submarine, there is no documentation saying so. As one Navy site says (emphasis mine):