I'm pretty sure this is already how the vault works. Unless I'm missing something. I did the same thing as the other comment state. It opens just fine without an internet connection.

----UPDATE----
Definitely does not work on desktop.
The desktop browser extension logs in to your account every time you open your browser.
This is a massive issue if your phone is not available. Even worse if it is and you have a massive password or secure note that's a pain to type in.

(I recently made another post about a self hosted option - ideally these all follow the same idea - we need offline access to our passwords - I shouldn't have to open my browser JUST to access a password. Many passwords and things are used offline or via LAN. (GPG keys, SSH keys, general notes, etc)

Does this not work already? I just put my phone in airplane mode and I was able to open Pass and see my vault.

Just wanted to emphasis about being able to have a local vault in the phone/computer to be able to access passwords offline.

The vault should be super heavily encrypted in the local unit and as soon as internet is available synced with your servers. This local vault might be an option and not default so people can store it locally if needed but keep it safe with you on your servers.
I just know when traveling the world I often end up without internet and still need something stored in the password manager like a note, card number, membership and other things I like to keeps in a safe place.

One could select parts that would be stored in the local vault. Like if I want my notes, Membership info to be available offline I could choose that in the app settings. The rest would be safely stored online safe on proton servers.