You may have more of a legal question, than a tax question. For example, what does the court order, giving the grandparent legal custody, say she gets to claim the dependent. Even if it does, there's some question whether it's even valid.
The general rule is the custodial parent has first priority on claiming the children on her taxes. The IRS goes by physical custody, not legal custody. The (physical) custodial parent does not need the legal custody taxpayer’s permission to claim the child.
Here’s the standard answer to that general question:
If someone else claimed your child inappropriately, and if they file first, your return will be rejected if e-filed. You would then need to file a return on paper, claiming the child as appropriate (including filing an amended return). The IRS will process your return and send you your refund, in the normal time. Shortly (up to a year) thereafter, you'll receive a letter from the IRS, stating that your child was claimed on another return. It will tell you that if you made a mistake to file an amended return and if you didn't make a mistake to do nothing. The other party will get the same letter you did. If one of you doesn't file an amended return, unclaiming the child, the next letter, from the IRS, will require you to provide proof. Be sure to reply in a timely manner.
Winner gets the tax benefits; loser gets to pay the IRS back with penalties and interest. The custodial parent almost always wins. The IRS goes by physical custody, not legal custody.
