Geez S., you are making assumptions based on your feelings. You don't know that she hasn't asked to meet you. You only know that your husband has not told you if she's asked.
You don't want to meet her even tho she is a friend of your husband and in-laws?
You don't even know her or her family. Why haven't you asked to meet them? This relationship seems to be having a major impact on your life even tho you know very little about both the relationship and the people involved.
We always fear most that which we don't know.
You can make this a family relationship by arranging for both families to get together. After you get to know her and her family you will have information that will help you decide how involved you want to be. This knowledge may or may not change your feelings about your husband and in-laws friendship with the woman. IT will give you information on which to base your decision. Men understand facts better than emotions.
I understand why he deletes her messages. He doesn't want to upset you. He might be more open if mentioning his friend didn't upset you. Your unwillingness to be a part of the relationship is dangerous for your own relationship with your husband. He most likely finds your attitude and non-acceptance of even getting to know her as jealousy when you may have no reason to be jealous. He may not know how to deal with your jealousy and anger. This could result in him withdrawing from you as he's withdrawn the e-mail. Who wants to be around anger and jealousy? When you were a teen didn't you not tell your parents about things that you knew were OK because your parents would be unhappy about them? Adults do the same thing. Keeping information about this friend from you could indicate that this relationship with her is inappropriate. But at this point in time it more likely indicates that he doesn't want to upset you. With the need to withhold important information may cause you both to begin fighting over this friend. You won't know if the difficulty in this marriage is caused by his outside relationship or because of your refusal to be a part of that relationship.
GET TO KNOW HER AND HER FAMILY. Have both husbands, both wives, and all kids get together. Or start with part of each family. I do suggest that the first meeting be within a group setting. Then you can decide if it would be possible for the families to socialize together. You will have clues as to her family's dynamics. You'll be better able to judge if she is a threat to your marriage.
For one partner to have a relationship with their ex outside of their relationship with ther current partner is dangerous. It is also unfair for you to try to end that relationship without knowing what that relationship actually is. If you do try to end the relationship it would be understandable if your rancor drove him away from you also. You would be controlling his life instead of being a part of his life. The two of you need to make a decision about this together.
You love your husband and he loves you. He chose this woman as his friend. It's quite possible that you would also like her if you gave her a chance. You both enjoy his company. I suggest that the 3 of you have similarities which encourage friendship.
Whatever happens after the two families get to know each other may result in either family not wanting to socialize. If you continue to feel that their relationship is inappropriate you'll then have actual information with which to discuss the situation. You will be better able to deal with reality base on experience instead of just your feelings.
Marriage should be based on trust. Your post sounds like you don't trust your husband. You may be right to not trust him but you will not know that you're right if you don't find more information.
As a police officer I rode often with male partners. One's whose marriages were solid arranged for me to meet their wives early in our assignment together. A part of a successful marriage is being open about your activities, friendships, and feelings. YOu've cut off communication with your husband regarding his friendship. If you don't re-establish communication you'll not be able to actually know what is happening. And you will continue to be upset which will also undermine your marriage.
I have had several men friends that were just that; men and friends. It is possible for friendship to happen between the opposite sex. We hear more about the affairs. In reality there are more friendships than unfaithfulness.
There is a quote that has helped me be more positive in my life. "We usually get what we're expecting." This is because we give energy to what we focus on. We spend time expecting the worst instead of looking for the best. We build on either expectation. Negative begets negative. Positive begets positive.