These avatars are only available annually, usually only 1 day! Here's a quick list for you to keep track of what day you need to get them.
I *heart* Sloth

Visit
(you need an account to see links) on
January 14. The answers are C D B C A.
Evil Jhudora

Complete one of
(you need an account to see links) on
February 4.
Valentine Chia

Send
(you need an account to see links) to someone on
February 13 OR 14.
Gadgadsbogen!

Visit
(you need an account to see links) on
March 4.
Illusen Doll

Complete one of
(you need an account to see links) on
March 17.
April Fools

Visit the news page (after the prank is taken down) on
April 1 (April Fools Day).
Easter Cybunny

Send
(you need an account to see links) to someone on
April 8.
Chef Bonju

Go to
(you need an account to see links) and mix together Blumaroo Steak, Orange Juice and Gourmet Cooking for Your Pet
anytime during August.
I recommend buying Gourmet Cooking for Your Pet BEFORE August because the book tends to inflate a TON.
Happy Birthday!

Visit the news page on
November 15.
Plastic Fir

Visit the Advent Calendar on
December 1.
Seasonal Attack Pea

Awarded on a
random day in December when you visit the Advent Calendar and collect your prize.