Piwik tracking, reporting, API doe not require access to the Internet, and will happily function without access to Internet.
However starting in Piwik 2.0, the new Marketplace feature calls the Marketplace API (hosted at plugins.piwik.org and themes.piwik.org). If there is no internet connection when Piwik tries to call the API, when you visit any page in the Admin of Piwik, you may experience the following error message:
curl_exec: Connection timed out after 3000 milliseconds or
curl_exec: couldn't connect to host or this error:
Warning: file_get_contents (http://plugins.piwik.org/api/1.0/plugins) failed to open stream: HTTP request failed!
To prevent this error, you may disable the “Marketplace” plugin in Admin > Plugins.
You can still download free plugins, as well as purchase premium plugins, directly on the Marketplace. After you download or purchase plugin(s) on the Marketplace, you can install them manually in your Piwik.