Can you create a new filegroup for a table in a database that is live?
You can definitely add a filegroup to a database while it's running, but it will have no data until you create a table in it, or rebuild a table onto the new filegroup.

This would mean dropping the clustered key on the table, and re-creating it on the filegroup:

CREATE CLUSTERED INDEX IK_Example ON Example ( id ) ON NewFileGroup;


OR

ALTER TABLE Example ADD CONSTRAINT PK_Example PRIMARY KEY CLUSTERED (id) ON NewFileGroup;


The Microsoft Docs on this one is pretty good.