Piwik can track visits on your intranet website as long as Piwik itself is hosted in your intranet. To make sure that all unique visitors are tracked correctly in your intranet, there are two choices:
1) Either make use of User ID feature to assign your users with a unique persistent User ID string. This will make sure your users are tracked accurately across all devices. Learn more in User ID guide.
2) Alternatively if you cannot use User ID then please edit your config/config.ini.php and add the following:
[Tracker] trust_visitors_cookies = 1
This will ensure that even if all visitors have the same IP, browser configuration, OS, etc. they will be counted as unique visitors based on the first party cookies. If not set, you will see a lower number of visits.
Note that this setting will also apply to your non intranet websites if you track in your Piwik both intranet and internet websites. We plan to differentiate between Websites and Intranets in a future version (more info).
If you use SharePoint for your intranet, check out our SharePoint Analytics product.