If I'm not mistaken, LastPass keeps both a local and cloud version of your password. When you're having trouble connecting to LastPass's servers, LastPass still functions. I think it's just that if the password changed on another device since the last time it synced locally, you'll still have the old one.
So I'm pretty sure that you'd have to lose basically every device it's on and have LastPass's cloud deleted to lose them all.
They don't keep a server version of the password (they don't have access to your password), but they do have the authentication credentials. A password manager that only worked online would be frustrating.
So I'm pretty sure that you'd have to lose basically every device it's on and have LastPass's cloud deleted to lose them all.