php-webp | WebP image read/write functions

So i've made a php script which allows me to download some specific files from another password-protected webpage using basic http-authentication, where I simply put the username and password in the URL like username:password@url.com. It works without problems, but i'm having trouble figuring out how to do the same thing on a webpage with cookie-session based authentication. In short, when a user goes to my PHP-webpage, they should be redirected to the login-form of the external webpage, then redirected back to my PHP-webpage. At this point my script should be able to communicate with the external webpage using the users credentials. What would be the easiest way to do this?

Thanks in advance!