Is it possible to play a multiplayer game on battle.net but with more than one computer in the same house? I want to play a 4 player game with my son and I at our house versus his friend and Dad at their house.

Why would you think this would be an issue? What limiting factors or technical issues are you worried about? Is there something special about your network archetecture that would cause an issue?

IPX IP connection support up to 4 players in Wc2, as i remember. You dont needed Bnet to local play.
But - as always - you need unique CD key\copy of the game to play online via official servers.

I was thinking that since I have to port forward to a static IP to be connected to battle.net that I wouldnt be able to use 2 pcs on the same local network and have them both connected to battle.net. Am I overthinking this? I just didnt want to buy multiple CD keys if its not going to work

It is an issue, but a solvable one. See my post in the Diablo forum about this setup - https://www.gog.com/forum/diablo/cannot_connect_to_bnet_after_successfully_portfowarding/post12 ; report back if you need more help. Despite being in the Diablo forum, it applies equally to Diablo, Warcraft 2, and Starcraft/Brood War (disc release; I never bought the remaster, so I cannot speak to how it works). They all use the same network setup.

you need unique CD key\copy of the game for official services.

Im following what you are saying in this post, but Im not sure that I know how to set my router up like that. I have a Netgear Nightawk AC1750 Model #R6700. I also have a subscription to express VPN. Would there be an easier way to connect via VPN on one of the local computers?

Port forwarding is required for hosting.

forward ports to the computer you want to host with, other computers who want to join don't matter.

https://us.battle.net/support/en/article/7842
Not needed visit any other sites except this one (tow - check link in the end of this article).

My experience was that the games would behave poorly, and sometimes very strangely, if players could not form a full mesh, letting every player send directly to every other player.Sorry, I know nothing about configuring those for this purpose.Maybe, but I would expect the VPN provider not to allow unsolicited inbound traffic, so you will have the same problem.

One way you might be able to test this without buying anything is to get everyone to install Blizzard's classic release of Starcraft, which was made a free download to promote their for-purchase remastered edition. I think the free-to-download classic Starcraft can play on battle.net (but I haven't found anything definitive on that point). If so, try to get everyone into a Starcraft game. If you succeed, your configuration should work for Warcraft 2 and you can go buy the extra keys you need. If you fail, then you know you need more work before you can play War2.