My sister and her husband have 3 children, two of them live with them and 1 lives with the husbands mom. They are moving out of their home 7 hours away and are leaving the kids with his mom who also does not have any income coming in and without going to court to give her custody or any right to take care of them. Could this count as child abandonment or anything illegal?

It is not illegal as long as you are not claiming the child is living with your for any social services, as long as medicaid, or other services the child may be getting knows who is raising the child and where the child is.

If you are claiming the child as if he is living with you, then that is illegal.

Of course the people who are raising the child, they could go to court to get guardianship or custody by showing they have been raising the child. They also could seek child support from the child's parents