Смысл в том, что у WebPlayer'a есть так называемый Security Sandbox. Загружая и отображая на экране текстуру, с не разрешенного места происходит блокировака чтения данных с экрана или рендертекстур. В результате скриншот не сделать.
http://docs.unity3d.com/Manual/SecuritySandbox.html
You are allowed to download images from servers that do not have a crossdomain.xml file. However, the only thing you are allowed to do with these images is use them as textures in your scene. You are not allowed to use GetPixel() on them. You are also not allowed to read back from the screen. Both attempts will result in a SecurityException being thrown:
SecurityException: No read access to the texture data:
at (wrapper managed-to-native) UnityEngine.Texture2D:GetPixel (int,int)
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