Hey there,

both the normal server and the community server are only available to players as long as the host is online. The difference is that the normal server is only visible to the host's Steam friends, while the community server is available to everyone that knows the password.

What you are looking for is a dedicated server. That will run the software continuously. You can either set one up yourself or rent one. Please use the forum search function, the Valheim wiki, or a search engine for the respective guides and recommendations.

Hope this helps, have a good day!