Does this game (Hot Brass) offer multiplayer coop play through LAN play?

Yes! The game mentions you can join other players in your LAN when you hover over the "Online" button in the menu. It's a little bit confusing, but I believe when you choose the "Online" mode, you can host/browse games that are on your LAN as well as in your chosen server region (e.g. USA West, USA East, Europe, etc.)

You can also play on one PC with 4 controllers, if that's a thing you want... though your characters are locked to only being a certain distance away from each other in this mode.

EDIT: There are options related to hosting LAN-only when you actually host a game... Image below:
https://i.imgur.com/3lrzNlO.png