My armatar is an example of such a seasonal editioned armatar, the 2009 summer edition to be exact.
But if you're talking about getting a picture of something other that these two options, for example having a picture of a real life cat as your armatar, then that's not possible.
Users have been able to circumvent this and use their own armatars here, but they were quickly slapped on the whrist and in some cases even banned for it if I remember correctly.
When the new ArmorGames v3.0 launches, a whole new 'world of armatars' will surely open up, maybe even with the former seasonal armatar sets included to them!
Whether or not they will be having contests for ArmorGames users to enter their own homemade armatars and add some sets created by an artist hired by AG is something that will depend on the community manager I think... and the big question is: "Will the be one in the future?"