The child support enforcement agency is now telling me that there was a "glitch" in their system and that there is a zero balance but a $5,000 interest payment due now.
Do I have a leg to stand on or am I going to have to pay it because of their mistake?
You certainly have a leg to stand on. You will need to show that a valid court order set the amount of child support arrearages you owed and that you have paid according to the order.
You will likely need to produce copies of both the order and any checks or other receipts showing the payments you made. If the agency is persistent in its pursuit of additional interest payments you may have to fight back. You shouldn't have to pay for their "glitch."
How Much Should You Be Paying?Child Support Calculators
You should seek the advice of an experienced family law attorney, such as the ones with Cordell & Cordell, in order to identify and evaluate your options for fighting the agency's attempts to saddle you with an additional $5,000 in debt after you've already paid what was required by the court.