First, are you eligible to claim your kids as dependents? The first place to look to determine this would be the Divorce or Custody Decree. Second, if after reviewing all the criteria you determine you are allowed to claim the kids on your taxes as dependents, then you should claim the them on your tax return; you'll probably have to do this by mail if they've already been claimed on another persons return. Then the IRS will require both parties claiming the kids as dependents to prove their claims. So, gather records and evidence to be ready for an audit. Lastly, if you are in the right and you are able to claim your kids as dependents then there will likely be no damages to sue for. Conversely, if you are not able to claim your kids as dependents based on the criteria, then, again there likely will be no damages to sue for. Simply, you probably have no case to sue your ex, there are other avenues for solving this scenario through the IRS.
As between parents, the custodial parent (the one who has physical custody more days out of the tax year is entitled, under federal tax law, to claim the children as dependents. The States courts have the power to award the right to claim the children in a divorce, custody or support case. Since you are paying child support, my assumption is that you are the non-custodial parent. If so, unless a court order has awarded you the right to claim them as dependents, you are not entitled to claim them no matter how much support you provide.
Because you are paying child support, I am assuming that the children do not live with you. Child support is generally paid to the parent who is caring for the children, and that is typically the custodial parent who lives with the children for most of the year.
One of the requirements that the I.R.S. has established for being able to claim a child as a qualifying dependent is that the child must live with you for more than half of the year. If your children do not reside with you for more than half of the year, then you can not claim them as dependents. As such, it is not your ex-spouse who is preventing you from claiming the children, but rather federal law that prevents you from claiming the children.
This residence test doesn't apply in some instances where the parents are divorced, but this only occurs when there is a divorce decree dictating who is able to claim the children. Without a divorce decree stating that you can claim the children, you won't have the ability to claim the children.
As far as your ex-spouse's return being fraudulent due to her allowing her parents to claim the children, that depends on whether the children live with her parents for more than half of the year and are financially supported by the parents. There are other requirements, of course, but ultimately if the parents satisfy those requirements, and there is no divorce decree stating that you can claim the children, then there's not much that you can do. If you do know that the parents don't meet the necessary requirements, then you may be able to report suspected tax fraud to the I.R.S., but you're not going to be able to personally sue them in a court of law, as you have no right to claim the children yourself, and thus are not incurring any sort of personal damages for which you could be compensated.