Hello fire772. The only way to accomplish what you are referring to is to have multiple browsers open at the same time. In each of the browsers you have listed with the exception of Internet Explorer, the vault will close each time your browser is closed. When your browser is used and the vault opened the vault should synch automatically from the cloud. Because I do not use FF I cannot vouch for the same using it.
One of your forum labels is 'Local Vault'. If in fact you do still have a local vault, the only browser it will work with is IE. All other browsers now work with independent browser extensions that only work with a cloud vault.
And it appears that there is no connection between the browser extensions. ie If you have the vault open on one browser, you still need to open it again in a second browser.