Gonna be real. This was a pain to get to work properly. The documentation is all over the place and there doesn't look to be one single way of how to get this working. I wrote a quick guide on how I got it to work on my end with play with my online buds, just using Hamachi.

Important Note: Complete all installation steps, regardless of whether you are hosting or joining. Tutorial made for Windows computers. Sorry mac and linux users yall are on your own.

1. Install Necessary Workshop Items
Subscribe to these two Postal 2 workshop items:

https://steamhost.cn/steamcommunity_com/sharedfiles/filedetails/?id=737631813
https://steamhost.cn/steamcommunity_com/sharedfiles/filedetails/?id=738015266

(honestly idk if the api is mandatory to install, but i just do it anyway)

On the Postal 2 library page:
Go to Options > Properties > Betas
Select beta 5025

Once the beta is installed, navigate to your Postal 2 local files:
Steam\steamapps\common\POSTAL2Complete\WorkshopContent
Delete both folders within the "WorkshopContent" folder (they should be named with random numbers starting with 73)

Open Postal 2 and allow the Nick's Co-op workshop items to install manually.
It is important that you and the people you’re playing with have the exact same install, and the exact same mods installed as well. It could cause issues if your installs are mismatched.
To FULLY uninstall mods for Postal 2, after unsubscribing from them on the Workshop, go to Options>Purge Workshop.

2. Set Up Hamachi
Download Hamachi from vpn.net.
Install Hamachi and create an account (free; disregard any script errors).
Create a network:
Network > Create a new network
Record the network name, password, and your IPv4 address (right-click on your IP in Hamachi's main window and copy).
Share the network name, IPv4, and password with those you want to play with.

3. Host a Game
Launch Postal 2 (ignore the Steam multiplayer prompt).
Start a new game.
Go to Workshop > Set game mode to Nick's Co-op.
Choose your difficulty and click Start.
After the game restarts, a new menu will appear.
Click "Co-op Game." (Optional: Change character textures and in-game name here).
Click "Host Game."
Set up server info:
Enter a server name, game password, and admin password.
Share these with your friends.
Click "New Game" to start the game on Monday or whatever map or gamemode you’ve all agreed on.

OR, IF NOT HOSTING/JOINING A GAME INSTEAD,

4. Join a Game
Join the host's Hamachi network.
In Postal 2, click "Co-op Game" > "Join Game."
Go to the "Direct" tab.
Paste the host's Hamachi IPv4 address.
Enter the game password when prompted.

This is fool-proof on my end. I'm not gonna respond to troubleshooting comments, i mostly just wanted to post this here just in case anyone else needed it. Have a postal day!!!!