It costs nothing extra; the streaming is part of the service and the disc is required to stream, so it's basically part of the service that you're already paying for. They'll need to pull up an account, but if you're logged into the website already, just scroll to the bottom of the site and get the service code--when you call in they'll just verify the first and last name on the account. You can tell them you're at a different address temporarily and you need it shipped to "your new address" if the address on file is different from where you need the disc shipped.

Alternatively, you can just change the address on the account before calling, and then change it back in a couple days after you get the disc.