<p>Do you know if it's possible to build an application for the LinkedIn platform?</p>

Yes, Linkedin has an API:
http://www.programmableweb.com/api/linkedin
http://blog.linkedin.com/blog/2007/12/the-intelligent.html
So you could build an application that uses it.
Update:
(
from second link
)
We’ll be phasing all of this in over the coming months and to get involved with the Intelligent Application Platform either for APIs, widgets, or hosted applications, send us an e-mail to
developers@linkedin.com
telling us what you want to build and what you need to build it.
Since there are published Mashups using LinkedIn I would assume that means you can use the API even if the documentation isn't readily available.
As a tip, in the future include links to what you found that didn't work, so we know not to give it to you again.
I poked around a bit more and I found some
more on their widgets
which appears to be the main focus of their API.