You can use Calibre across the network. I do so with a Windows 7 desktop and a Windows 10 laptop and a Windows 10 Surface Pro. I have the laptop and the Surface mapping the external drive on the desktop and I connect to that. But I only connect one at a time so I don't accidentally connect to the same library more than once.
You have to use password protected networking. If the Vista system does not get logged in then you'll need to create an account to use for networking. if you do log in, just use that account. You map the external drive from the laptop. Once the network link is setup, you can just access your Calibre Libraries no bother. But before you do, make sure you have full read/write access to the drive.
Another issue is that you have a serious problem with the Vista system. The external drive is not needed to run the system and because it is, something is very wrong that needs to be fixed. Something in Vista is accessing the external drive in a way it should not be. Check to see what the Vista system is doing with the external drive. You should be able to disconnect the external drive and the system boot no problem. One thing to try is disconnect the external drive and try booting in safe mode and see if it works. If it does work, then something that's running at startup is causing a problem and you can just not run it or fix it so it doesnt need the external drive ever. But this is an issue that needs fixing because that's not the way Windows is supposed to work. Something is seriously broken.
Read the following in case it's your last ditch effort to fix the Vista system.
https://www.bleepingcomputer.com/tut...tartup-repair/