Is that HTTP or HTTPS? In the former case, it is just a matter of using POST request to send the right credentials. In the latter case, it can be harder...

Well, to my knowledge, Processing doesn't do HTTPS.
Java can probably do that, but you have to search a bit the Web for such information.

I just stumbled upon an example of making an HTTPS connection to a server.
I rewrote it a bit, you can find my version at: http://bazaar.launchpad.net/~philho/+junk/Java/annotate/head:/_QuickExperiments/ReadHTTPS.java
That's pure Java code but adapting it to Processing shouldn't be hard.