I made a button just like that: http://www.hitwake.com/inbound-button/. An API would've been useful, but got by without one.
Hit me up on Twitter or somewhere if you ever start with an API and want to hear my insights. Basically just a JSONP endpoint to fetch the counter would probably be enough.
Here's a new inbound.org button I made: http://www.hitwake.com/inbound-button/
There's still no API, and the button may (will?) break if the inbound.org site is changed in certain places.
Great! Head over to login and use your credentials there