I started using Google Chrome as the main browser, but I skipped the Evernote extension, which can pin a web page directly to the local Evernote application. Is it possible for me to write an extension in Chrome that can do this?
Yes, it is possible through NPAPI, but your local application should be prepared for external communication. The code running in the NPAPI plugin has full permissions of the current user and is not isolated or protected from malicious input by Google Chrome.
All of this is described here: http://code.google.com/chrome/extensions/npapi.html
To avoid the NPAPI method, another idea would be to communicate with a custom local HTTP server attached to localhost and send requests to it.
Disclaimer: never tried, but theoretically it should work.
I don’t think that chrome allows this, simply because it would be dangerous for plugins to have extended privileges, they even run in an additional stream with low access rights and only communicate with chrome itself through channels.