I will just spill out few ideas here, and you see what you can work from that
1. Additional router
You could buy an additional router, which you would change to client mode (it would connect to gopro), and connect it to GoPro. Then, that router would be connected to your existing router, thus giving you connection from your computer to the camera.
Bad side of this is obviously buying a new router. On the bright side, router would probably automatically connect when the camera is in range and you would have instant access.
2. Current router
If you are not using your current router's wi-fi (highly unlikely, but possible), you could change its mode of operation to client, and work same as no (1)
3. USB Wi-Fi for your computer
As already pointed out above, you can buy cheap USB Wi-Fi adapter for the computer.
4. Removable disk over the LAN
If I read right, when you connect your GoPro to the computer, you get the removable disk from which you can copy files? If so, then you could find a router (or maybe your current router has an option) to which you could connect GoPro using USB and share its storage with everyone on network.
Downside of this is that GoPro probably cannot work while connected to USB, which means you will probably be unable to record while connected, although I am not sure about that. Also, this is basically the same as connecting GoPro to the computer (i.e. is not wireless).
This all is only my opinion, and I don't know if any of these ideas are applicable to your case.
Let me know what you think.