If there is a court order that says he has to pay half of the medical expenses and daycare costs you can check with them to see if they will collect them for you if you submit the receipts to them.
If it's in the court order, you can always file contempt of court against your ex compelling him to pay.
How much more do you make than him? If you are going by what the two of you were making at the time of divorce, that may no longer be accurate.
Anyway, if it isn't that much more than him, he may end up paying you child support if you are the one who is paying for health insurance and childcare costs. Those amounts are add-ons to child support.
Who has the tax exemption? And is it tied into him being current on all of his financial obligations to you? If not, I would file for a modification to add that to the court order. That alone may be an incentive to him to keep current.
Personally, I would consult with an attorney, s/he should know if the agency that collects child support will collect for the medical expenses.