This is a family law matter, rather than a tax matter. Contact the family law attorney that represented you in your divorce and discuss your options.
Any individual seeking legal advice for their own situation should retain their own legal counsel as this response provides information that is general in nature and not specific to any person's unique situation. Circular 230 Disclaimer - Advice given in this response cannot be used to eliminate penalties with the IRS or any other governmental agency.
If your agreement is in writing and was made part of the divorce, your ex may be in violation of a court order if he refuses to give you your half of the tax refund. You could ask the court to hold a show cause hearing and he will have to explain why he should not be held in contempt of court, which is serious and can result in jail time. If your agreement was not in writing, you are probably out of luck. If the children really reside with you, and you do not have a written agreement or court order allowing your ex to claim the children, you should be entitled to claim the children. It is always wise to consult with a tax attorney or a CPA prior to filing tax returns or making any decisions that have potential consequences.
This answer is provided for informational purposes only. It is not legal advice and is not intended to form an attorney-client relationship.