Help I have exactly the same problem but uninstalling, emptying trash and restarting has not fixed mine.
I have access to the vault via the app but not on Safari, our last pass is part of a business account and not being able to use it on my Mac mini, M1, Big Sur is a disaster for me!
Could you try these steps to uninstall the LastPass Mac app, restart your computer, then reinstall?
While the Lastpass folks get their native app up and running for safari you can use chrome. Google already has a native browser that runs on the M1 chip and ironically the Lastlass extension runs just fine. I am not sure why it runs on Chrome and not Safari but its an option if you are a Chrome user.