If you downloaded and installed the .rpm directly then it will not receive automatic updates, you’d have to keep an eye on their website and manually install new updates. You should avoid installing software like this wherever possible. To install it properly, you’ll need to add the mullvad repository and then layer the mullvad-vpn package via rpm-ostree, this way it will be updated automatically along with the rest of your system. Because bazzite is based on fedora-atomic, adding repositories ...
SkatorGaytor describes the steps to install Mullvad VPN on Bazzite in an answer on this thread (post 2).