Partly. There is no LastPass program to install on Linux, but the browser extension and the website both work in Firefox and Chrome on Linux, and your vault is all there.
LastPass by LastPass · Security & passwords · Works in a browser
LastPass publishes a desktop app for Windows and Mac. Its download page offers Linux users the browser
extension instead, for Firefox, Chrome and Opera.
In practice that is most of LastPass. The extension fills in logins, saves new ones and opens your vault, and
the website covers the rest. What you lose is filling passwords into programs that are not browsers, which the
Windows desktop app could do.
Nothing happens to your saved passwords. They live in the LastPass account, not on the Windows machine, so
signing in on the Linux side brings everything with you.
It is also a fair moment to reconsider. If you were ever going to change password managers, doing it while you
are already changing operating system saves doing the export and the clean-up twice. Bitwarden and KeePassXC
both have real Linux apps rather than an extension only.