The specific application that launches when you click "Update Status" is called gwibber-poster. The actual application that handles the Broadcast accounts in Ubuntu is (by default) Gwibber.
To give it a keyboard shortcut, simply open up System Settings -> Keyboard -> Shortcuts, and create a custom shortcut for the command gwibber-poster. I don't have enough rep to post images yet, but here's a link to the image so you know what I'm talking about: Custom Shortcut
Funny enough, I actually figured this out because I was looking up the exact same question. I knew the application was Gwibber, I just didn't know whether there was an option or another command that launched specifically that "Update Status" dialog. So, I typed "gwibber" into a terminal window and hit Tab until it showed me the available commands that started with "gwibber". That's when I saw "gwibber-poster" and decided to give it a shot. Glad I found out about this. Hope it helps somebody else!