Doing some testing with deployed package now, the sandbox doesn’t seem to be a problem for me, but I do get different results from the key, I have an idea, not sure if I can do what I am thinking yet, but if I can, I will post the package for you to try or at the very least, the code.
Ok, one snag:
Texture2D.LoadImage.html
That will only handle JPG and PNG files, the file you are trying to grab is a GIF, trying to find a way to load in a GIF by byte array at the moment, time for lunch.
Edit: 12/17/2010 13:07pm 
Since Unity texture loader is looking for the PNG or JPG and then we are talking about grabbing avatars from Facebook (or any other system for that matter), the thought crossed my mind, on the server you are hosting your game, why not create a form that you can use to translate other image times to PNG or JPG? So thinking this through, Unity in its little sandbox can in fact send POST and GET type commands, ok, so where do we sit? We have a URL that will actually return the BYTEs of our image, the header of that is our image type with the contents containing the image data. Well, in theory we should be able to accomplish the following, create either a php script or asp script or even asp.net script (slowest of the bunch) that has a web form that you can post to, giving it the URL that we know we want (not the final URL), then that script goes out and retrieves the byte array of the image, so far so good, now that byte array then is converted via your code on that web form from a GIF to JPG or PNG, this results in another set of bytes that is valid for Unity to understand.
With me so far? Good… Now, on that final round type, we give Unity the converted image bytes OR to streamline this even more, create a MySQL database and a table in it that holds the images, first, you have a query that looks to see if the requested URL already exists on the table, if it does, then return the image to Unity, if it does not, go out, get the image, convert the image, store the JPG or PNG in your database and return that to Unity, so you are prefetching and caching the image to avoid the trip to Facebook… Still working out the issues since to do this mechanically, you need to use System.Drawing.DLL and no DLL’s can be pushed out within Unity’s sandbox, so you have to dance a little dance.