My son has been told for about 9months he is the father. Baby is due any day now her mom won't let us know anything. Can the court order a DNA test on newborn at time of delivery at my sons expense .

The court can, but you will have to request it. Depending on where you live, most hospitals don't do the DNA test, and it will be done weeks/months after delivery. Also, dependent on where you live, there may be a putative father's registry that your son can get on.

Your son( not you) will have to file in court to prove paternity, I doubt it will have a court order ready the day of birth, but may take some time to get the order heard and signed by judge.

Where - it matters.