Can a parent be deportated again if child support is in process?

Seeking and obtaining a child support order against a non-citizen would not in and of itself cause a deportation.

I would need the details, however, generally, there is little coordination between the family courts and the immigration service on support matters.

The Federal Government has no interest in subordinating national immigration regulation to your private financial needs.

This is a matter which an immigration lawyer can comment on much better than any other kind of lawyer. I strongly suggest you direct your question to such a lawyer. Immigration law is fairly complicated, and is its own specialty. As a matter of mere impression, it would seem to me that the family court or child support agency would not want him deported: their purpose is much more likely to be forcing him to pay child support, which is harder to do if he is outside the US. The outstanding warrants are troubling depending on what accusations of wrongdoing they are based on, they might lead to his being jailed and unable to work anyway. Good Luck.

Immigration does not care about support and seems to function independently. He will likely get deported and you can probably proceed and get a support order, but collecting it internationally will be almost impossible.