The only parent who has the automatic right to claim a child as a dependent is the parent where the child physically lived more than half the nights of the year. That parent gets to claim the child tax credit, qualify for head of household status, EIC and the dependent care credit. You have to count the number of nights -- "custodial" parent based on a decree or order has no effect, the IRS follows federal law which supersedes any state court matter. That is what the IRS calls custodial parent, counting the nights where the child slept.
If that was the other parent, then you can't claim the child this year.
The only way the non-custodial parent can claim the child as a dependent is with a signed form 8332 dependent waiver, or a pre-2009 divorce decree that says certain specific things. Using the waiver or the decree, the non-custodial parent can claim the child tax credit, and the dependent exemption (for 2017 and earlier). Head of household, EIC and the dependent care credit can never be waived, shared or transferred.
If the child lived somewhere else (not with the other parent) so that neither parent had custody more than 183 nights, things get tricky, ask here for more help. But if the child lived with the mom for at least 183 or more nights, regardless of any order, and regardless of where the child lived during the day, that is the person who can claim the child as a dependent this year.
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Now, regarding your pre-2009 court order. The non-custodial parent can claim the child using the court order if it says 3 specific things.
The noncustodial parent can claim the child as a dependent without regard to any condition, such as payment of support.
The custodial parent will not claim the child as a dependent for the year.
The years for which the noncustodial parent, rather than the custodial parent, can claim the child as a dependent.
In other words, it has to say what years Dad can claim the child, it has to say Mom won't claim the child in those years, and there can't be any conditions. (It's not the IRS job to verify if child support is up to date or if Dad made the right number of visits to qualify.) If your order is at all vague on any of those three points, it doesn't count.
See page 11 here https://www.irs.gov/pub/irs-pdf/p504.pdf
If, based on the order, he could have claimed the child and didn't, her certainly could file an amended return for those years. As of today, it is only possible to amend 2015, 2016 and 2017. 2014 and older are too stale to amend.
When filing the amended return, he must attach a copy of the order (not the whole thing) with the pages that discuss the tax treatment, the cover page, and the signature page. The IRS will contact his ex and investigate, and if they find in your favor, she may have to repay certain items with interest.
