You can save the image on your server and then pass the image path to your Email Service.
public static void SaveImageFromUrl(string imageUrl)
{
WebRequest request = WebRequest.Create(imageUrl);
request.Method = "GET";
WebResponse response = null;
try
{
response = request.GetResponse();
}
catch (Exception Ex)
{
}
Console.WriteLine(((HttpWebResponse)response).StatusDescription);
Stream dataStream = response.GetResponseStream();
StreamReader reader = new StreamReader(dataStream);
string ImagePath = HttpContext.Current.Server.MapPath("~/attachments/") + Guid.NewGuid() + ".jpg";
using (Stream s = File.Create(ImagePath))
{
dataStream.CopyTo(s);
}
}