He hasn't paid any child support in over 6 months. He wants to claim her on taxes this year.

Only the custodial parent can claim the child. It does not matter if child support is paid or not. Your U.S. Tax Return: Child of separated or divorced parents

Hold on! There may be a separation agreement which dictates who gets to claim the child. Such an agreement is a legally-binding contract which the IRS will uphold if made aware of it.

Normally, the custodial parent holds all the cards in this kind of situation, but NOT always.