They may just toss the letter after they receive it. If /u/DrD3adpool still knows someone in that town he can ask them to tell his family that he wants to contact them and ask them to oblige. After that the only option is showing up in person.
In subreddit r/RedditForGrownups
In a post titled:
I've been the black sheep of my family for years and actually haven't had much contact with them for five years. How to reestablish connection with family?