Need to include FKs in SQL Indexed View clustered index?
SQL Server will always be able to automatically update the view's index(es) as the source table(s) change; there's nothing extra you need to do.

You can inspect a query plan that modifies one of the source tables, you'll see there are extra operators to update the view's index(es), the same way it has to update all the applicable nonclustered indexes on the source tables when data changes.