Is the word "yearling" appropriate for a recurring event?

As adjective, the NOAD defines yearling as having lived or existed for a year; a year old.
Dictionary.com defines the adjective as meaning of a year's duration or standing.

For recurrent events, yearly is the word I would use. If you interpret existed for a year as using a relative reference, then yearling could be used to describe the badge.

A yearling is a term which I'd associate with horses. I assumed that the stackoverflow usage was a joke rather than a serious name. Why not just call it the "two/three/four year" badge or just give two, three or four yearling badges.

As others have pointed out, yearling does not have a use for recurring events. Other than on StackOverflow.com, it seems to only be used (in American English) when referring to the age of animals, especially horses. For a recurring event, the word you should probably use is either annual or anniversary, or yearly. As a native speaker of American English, I am much more likely to use "annually" than "yearly".

It would not, however, surprise me, if StackOverflow continued with the humor and referred to users who have been participating for two to three years as toddlers.