Access to the cookie with an another webapp

Is it possible to write a webapp which can use an another webapp's cookies, sessions, etc? I know I can write a C# or other desktop application which can it do.

I know generally it isn't possible but always there is a little hole in the security.

For instance, I log in my Facebook account in the browser and then I close the tab. After a few minutes, I reopen the browser and then the Facebook and I still signed in.

Can I write a webapp which for example open the Facebook and write a post?