If someone is behind on child support payments can the bank refuse a loan for a car?

Its possible but no most consider it and don't always act on it. There is no law affecting it to go either way.

It affects your credit. Most likely they will if it is a substantial amount and they know your wages are going to be heavily garnished. If it is really late and not being paid at all, they consider the possibility that you may end up in jail and they won't receive their payment. The amount owed and a steady job would be big factors.

If you are caught up now, write to the State to make sure it shows that you are current so that your credit score can improve.