Most screenshot services do not support this, however we have opened up the ability to set cookies in GrabzIt therefore if your website login uses sessions, you could set the user session cookie information.
Sessions like the ones used in PHP and ASP.NET etc, store the user's data on the server. The server identifies who the session data belongs to by putting a session cookie on the user's browser. Therefore if you assign that session cookie to GrabzIt all the session data will be available to GrabzIt when the screenshot is taken.
An example of this using PHP is shown below.
$sessionValue = $_COOKIE['PHPSESSID'];
$grabzIt->SetCookie('PHPSESSID', 'example.com', $sessionValue);
This example assumes that the PHP session cookie is named PHPSESSID.
Of course this is a complicated subject and so might be difficult to implement. Therefore you could attempt to debug any cookie issues, by signing into the target website and use the browsers built in developer tools, to do this in the Chrome browser just press F12. Then identify the websites session cookie and add name, domain and value to GrabzIt by using the custom cookies page, it is a good idea to use a expiration date a long time in the future to ensure the session cookie is not deleted.
We also support taking screenshots on a webpages that use basic authentication.