As others have said - do not use FTP - there are many, many things wrong with it. OTOH, you should probably use a protocol which does not require users to install additional client side software; although ssh based access (sftp, scp) would be ideal, MSWindows machines do not come with a client as standard.
There are no end of file managers available written in PHP (and probably other languages too). IIS does some very odd things with tunnelling of NTLM permissions - but if you're using Apache it should be reasonably safe - however accessing the drives is not as simple as just reading from \server\username\ - you'd need to allow the webserver admin rights on these files - but that's a security nightmare!
The way I do it (for a very different type of application) is to handle the I/O via smbclient using the username/password from the users PHP session - but that's on a Linux machine - AFAIK, there is no equivalent on MSWindows.
Certainly you would want to use the credentials supplied by the user to authenticate against the storage.
This is fast becoming scary complicated!
So a practical solution would be to allow the students to map the drives across the internet. That way you don't need to worry about the complications of providing a file manager which is not an integral part of the operating system (i.e. directly accessible from applications). But you don't want to expose your server to anyone and everyone. The usual way to solve this problem would be a VPN - and there are many good and low-cost ones available. Or simply wrap the service in SSL with client certificate authentication. But again this requires client side installations.
How about a web form to authenticate a user at an IP address which then creates a time-limited hole in the firewall for that IP address to connect via SMB (and they still need to provide a username/password to map the drive of course).