In Laws

My husband currently has no contact with his family. They live across the country from us. After a really bad divorce he lost contact with them and continuously blames it on them. However, I saw an email from his sister asking why he won't talk to her and asking what she did wrong. His parents helped him financially in his time of need through the divorce. He now claims none of them will talk to him because they all just wanted his money. However, after seeing emails from his sister and parents, I don't believe this at all. 
His family does not have any social media accounts but I have their email addresses. Would it be bad if I reached out to his sister and introduced myself? I want to have a relationship with them as family is very important to me. I'm currently pregnant with our second child I want our children to have a relationship with them as well. 
He has another daughter from his previous marriage that was allowed to skype and visit with them often. I don't think it's fair that my children don't know their other grandparents and aunts/uncles and she does.  Any advice?