If you were going through the same experience this family has, wouldn't you want others to know about it and help you in any way they could? I'm sure you would. And yet as far as you know, this family has had the added burden of facing this crisis alone.
This is why I encourage you to reach out to them and let them know you care. Yes, perhaps they are very private and don't want help from others–but how will you know unless you try? They may just be shy, or feel that the neighborhood hasn't been very welcoming. Remember the familiar words of Jesus (which we often call "the Golden Rule"): "In everything, do to others what you would have them do to you" (Matthew 7:12).
One way to reach out would be to do something practical for them–bake a cake, or take them a casserole or a pot of homemade soup. (Over the years, I suspect my wife, Ruth, has provided hundreds of meals to neighbors who were going through some kind of difficulty.) Or you might offer to go shopping for them or go with her to the doctor, or do something else they find difficult right now.
Most of all, let them know you are praying for them–and be sure you do. Remember: God may want to use you to point them to the greatest friend anyone can ever have: the Lord Jesus Christ.