Do you know if it's possible to build an application for the LinkedIn platform?
While LinkedIn has promised a public API for a very long time now, they have yet to deliver.
No, there is no public LinkedIn API yet.
IMO, their widgets (which there are only two of at the moment, which are very limited) don't count.
They say that they are open to being contacted with specific uses for their API and they may give access to parts as needed - but that is if they accept your ideas for integration. They have been very picky with this - and have not accepted my attempts to integrate with LinkedIn yet, they tell me I have to wait with everyone else, apparently my applications are not "high-profile" enough.
Sure, you'll find many Google results talking about their "promised" API, but they are empty promises and won't be of much help.