Maybe his mom doesn't know who the father is -- like, she really didn't know him. Maybe she was raped. Maybe she won't talk about it because she doesn't want her kid to carry that stigma. Maybe a hundred other things.
I guess you could contact these potential cousins and ask if they know of a relative who lived in X city around the time your husband was born.
What would you do differently if you had a full medical history?
Is the man that your kids call grandpa the man who raised your husband? Is he the man that your mother in law is married to? What led your husband to suspect that he wasn't his actual father?