[See http://rhizohm.net/irhetoric/post/2013/05/01/Getting-Twitter-Avatar-Via-C-With-11-API.aspx for how to do this with the Twitter 1.1 API]
Until Twitter deprecates their version 1 api, getting an avatar from Twitter anonymously serverside with C# is a snap. A little goofy since it returns a 302 so you have to set AllowAutoRedirect to false and then look in the Header location to get the URI:
private string GetTwitterAvatarUrl(string twitterHandle)  
            string avatarUrl = string.Empty;
            HttpWebRequest request = (HttpWebRequest)WebRequest.Create(string.Format("https://api.twitter.com/1/users/profile_image?screen_name={0}";, twitterHandle));
            request.AllowAutoRedirect = false;
            request.Method = "GET";
                WebResponse response = request.GetResponse();
                if (response.Headers["Location"] != null)
                    avatarUrl = response.Headers["Location"];
            catch (WebException) {}

            return avatarUrl;