Can I play Call of Duty: Black Ops III on virtual LAN?

Looks like it, Steam itself provides a very comprehensive guide on how to set it up: https://steamcommunity.com/sharedfiles/filedetails/?id=814090509

Long story short:

• In Steam, find Call of Duty: Black Ops III - Dedicated Server and Call of Duty: Black Ops III - Remote Console (the latter is the admining tool)
• Server files can be found in your game dir, usually C:\Program Files (x86)\Steam\steamapps\common\Call of Duty Black Ops III
• There will be a Launch_Server.bat which contains a few startup variables that you will want to change (map playlist, server name, password etc)
• The playlist itself is another file of interest (map rotation, gamemodes etc), in machinecfg\playlists.info
• There is of course way more, but these are the basics.

And a couple of notes:

• Although you can probably run this separately on your gaming machine, this works best on a separate PC. The rcon (admin) tool that you installed from Steam makes it possible to remotely manage it once it's running (changing maps, kicking unwanted players etc). The server PC can be a little older (a regular server will require fewer resources than the full client), but will also need Steam.
• You can set a password using +set live_steam_server_password "password", which would keep people out if you just open it up to the internet (which would also involve port forwarding on your router). For security reasons I would definitely run it on that VLAN that you share with your friends though.
• Always make sure you have a backup of a working configuration (Launch_Server.bat, the playlists in the machinecfg dir etc). You don't need to copy the full dir, just the config files (and preferably keep a few versions back). A typo in these files can prevent the server from starting, this is an easy fallback option.