First, I don't have much experience with websocket.
In theory, if a web app can use sqlite, it should be able to use other database engines like MySQL, MS SQL or PostgreSQL.
As commented by MicroDrie above, developers just need to find the differences such as the library, driver and SQL syntax in order to do the migration.
I have built
web api server which can connect to either SQLite or MySQL. So to answer your question, it is possible and it may not difficult as you have expected.