Yes, it's possible to run a website on a private network, though the "via a network drive" part of your requirement isn't clear. A website requires a web server (like WAMP or XAMPP or similar); you access this website via a URL; the URL can be embedded in a shortcut which resides on your network drive, if that's what makes you happy. The web server software can run on the same physical hardware as your network drive, but it's probably better if it runs on a machine of its own.
If you know how to build a secure website, it's probably safer than leaving an access database on a network drive. If you don't know how to build a secure website, it may well still be safer than leaving an access database on the network drive, as long as the website can never be reached from outside your corporate network, and your corporate network has sufficient security measures built in. If there's any way for an attacker to get access to your web page, and your site contains security vulnerabilities, the likelihood is that your data will get stolen.