Can my mother in law take our kids if we, parents, move out of state?

This is a little weird. If you have an agreement with your mother-in-law (presumably your husband's mother) to care for your children in her home so that they can finish their school year, it would be very difficult for her to assert that you have abandoned your children. The best thing to do would be to put together a written agreement which would include a provision that you reimburse your mother-in-law for any specific expenses she would incur while you are gone (Doctor visits, clothing, school expenses, etc.). One additional thing you should do is to inform the school that you are moving to a specific location with contact information and that you still want to be informed about anything relating to your children and that your mother-in-law is an interim caretaker, not a replacement parent.