When I open BW with the extension, it goes to Tab. How to open each time to Vault instead?
Not sure what you mean by Tab, but to open the Web Vault you need to point your browser at https://vault.bitwarden.com/
The various ways of accessing Bitwarden are (largely) separate, they are accessing the same data but different applications.
Why would you want that? The vault contains all items, which can be a huge list. Tab on the other hand only shows those items that match the current URL. If you do not get any hits under Tab then make sure to add URLs to your items.
Extensions shows , so I have an item with a matching URL.
Clicking the extension shows:
So I can either select this or - which is so much faster - press Ctrl + Shift + L to use those credentials to log into this forum.