This is probably a good time to talk to your daughter about how past indiscretions come back to haunt us. It's also a good time to talk to your husband about your MIL's memory issues. But before you do this, I would consider getting an appointment with a counselor who works with blended families. Your husband needs to actually see this child as HIS daughter, not just your daughter.
I think that you and your husband are both a little too singular in your thoughts here, each one of you on the opposite poles. And a lot of it probably because she's not "his" child. That's why I think you two need some counseling.
If she indeed has a problem with stealing, the only way to help her is to manage this issue on the same page, and not the way you two are currently thinking. If she ISN'T stealing, handling this wrong could cause her to start, or act out in a different way. If she is, handling it wrong could teach her to manipulate you.
I don't know how much money this is that your MIL is talking about, but if it's not a significant amount, your MIL is thinking the same way - that this is not her grandchild, but your kid. Nothing is going to change that, but if she didn't do this, her grandmother's accusations will make her feel very unloved. Please get help with this.