short answer: no
but you can play private matches online with your friends
##
Edit:
Or you play with your friend using the splitscreen feature.
one player plays with mouse + keyboard, the other player needs a gamepad to play the local coop (limited to 2 players)
Notice:
Some websites tells you, that the game has lan coop (e.g.

)
Quote:
Call of Duty: Black Ops III Co-Op Informationen (PC)
Local Co-Op 2 Players
Local + Online Co-Op not supported
Online Co-Op 4 Players
Lan / System-Link Co-Op 4 Players
|
but i cant see any option to play through lan...