How about a few differents categories? That way there is not just one winner. It would be lots of fun to see animals win in different categories.

A few ideas:

Photo contests - biggest, smallest, goofiest, most original, longest ears, best friends, bravest, tricks, etc.

Pet stories - share a story about your pet (rescue, cute things, adventures, etc.)

Tricks - most original trick your pet does

Etc., etc., etc. - I could think of a ton, but I'll spare you all the agony.

I love the idea of having a contest with multiple categories. Keep me informed on what's going on.