Would it be proper for a concerned friend or relative to privately baptize a child whose parents, though raised in the Lutheran faith, have been negligent in doing so?
Our normal practice is not to baptize children without the consent of their parents, unless the child’s death is imminent. That practice recognizes that parents have primary responsibility for the spiritual care and upbringing of their children (Ephesians 6:4).
I would encourage a conversation with the parents. If they were raised “in the Lutheran faith” and presumably baptized as infants themselves, ask them how their thinking and beliefs have changed since then. Their answers and responses will tell you how to proceed. I also wonder if it is possible for one of our pastors to visit them and speak with them about Baptism.
Your question about Baptism underscores the importance of that sacrament, and yet we need to keep it in perspective. While the Holy Spirit can plant saving faith in hearts through Baptism, that faith needs to be nurtured through regular use of God’s word. That calls for even more parental involvement in the lives of their children.
Do what you can to encourage these parents to carry out their parental responsibilities in having their child baptized. It goes without saying that this includes praying for them. God bless your efforts.