So, I was looking around on how to choose an avatar, and as far as I can tell, you can only choose one from an image on this image site.

The problem being: I'd like one of Polar Bear from Shirokuma Cafe, but there are literally no images on this very site of him. So I was basically wondering if I could have one of the moderators change it for me if I sent them the avatar image in question? That way, I'd have the avatar I wanted, and the staff can make sure I won't post anything that breaks site rule. (Even if I wasn't going to do that in the first place anyway.)

No. Even I wouldn't ask for that.