This child has obvious behavioural problems, and tackling the stealing will address the symptoms, but may not help the cause...he's doing this as a cry for help.
I think you need to have a long talk with your husband about this, on neutral territory, away from the boy.
Reassure your husband of two things: One you love him, and want to love his son just as much, but that this is causing problems, and they won't go away by disharmony or not discussing the issue.
Two:
You are convinced the child is desperately sad, and that he needs help and support. Perhaps the reason he has no problem at his mom's house, and that she refuses to help, is actually where the problem lies.... ?
You also both need to present a united front to the boy, but in such a way that there is comfort and reassurance in your attitude to him, not resentment and accusation.
Be in no doubt. he knows it's wrong, but he can't stop himself. To my way of thinking, this problem goes a lot deeper than "I want it so I'm going to have it".
There is the other point that he might need professional counselling. This may not be a situation either of you can tackle on your own, and if that's the case, there's no need that you should.
Many blessings.
let us know how you get on.