I need a way to send the player's username & password from my website to the game after they login.
What you are looking for is an authentication token. Your question isn't actually all that different from mine, actually: construct.net/en/forum/construct-3/how-do-i-8/user-authentication-custom-157301
If you log in and store the token on the website, it should be accessible as a cookie, assuming the game's page is also within that same website; you can use JavaScript to grab it most likely, then just store it as a global variable.
And just to also drive this point home too...
User authentication is an extremely tricky subject, though I agree that using cookies and session data within Construct 3 is not easy because no tools are provided to do so without the use of Javascript.