I was wondering if there is an API available for the "Open with" / share dialog. I see 3 social networks (Twitter, Facebook, and a JPN site) by default in iOS, but I was wondering if it was possible to register your own app to accept these options.
It's not quite clear what you're asking. If you just want to present that Open With dialog, it's the UIActivityViewController. You can even implement your own custom UIActivity to appear in the dialog, but that will only appear if the user summons the activity view in your app.
http://www.apeth.com/iOSBook/ch26.html#_activity_view
If you're asking how your app can appear when the activity view is summoned in another app, if you export a document type in your Info.plist, your app will appear in the Open With dialog if the user starts with a document of that type.