Is it permissible, Islamically, to return gold that was taken and sold secretly without the family's knowledge, or is it obligatory to inform them and seek their permission first?

It is sufficient to return the stolen gold along with sincere repentance. It is not prescribed to inform anyone about what happened, as long as Allah has concealed it and you have decided to return the stolen item. Rather, what is prescribed is to seek concealment under Allah's cover.