Is a person held accountable for the sins of his family committed via the internet he provided for them, despite his personal commitment to using the internet only for what is lawful?

There is no blame upon you for allowing family members to use the internet as long as you do not know that they are using it for forbidden things. There is no sin upon you if they use it for something forbidden without your knowledge. It is recommended to remind them of God's watchful eye and to avoid using it for forbidden purposes.