What does the custody and visitation agreement say? Some agreements include a clause that if the custodial parent is unable to care for the child for an extended period of time (which is often spelled out in the agreement...maybe 8 hours, 12 hours, 24 hours, whatever) then the other parent has the opportunity to care for the child during that time if he or she so chooses. So basically, non-custodial parent instead of a baby-sitter. This does not mean that you husband has to be glued to his son 24/7.
My guess is that there is no such clause in the agreement. If that's the case, then she has to first petition the court to get a modification. The police do not come and take kids away from their parents. The police only get involved if a parent is in violation of the court ordered agreement and all other avenues have been exhausted, or there is an emergency. In either case, they don't just go and act on the whims of the complaining parent, there needs to be some kind of order authorizing intervention.
ETA you need a custody agreement YESTERDAY. Assuming that they were never married, he probably didn't establish paternity? If he didn't he currently has no parental rights at all because in NC, an unwed father has no rights until paternity has been established. That said, the police still won't come and take his son away from him without some kind of court order. He really needs to establish paternity and get a formal custody agreement. Paternity can be established via the filing of an Affidavit of Parentage form, but I believe that this requires the mother's signature as well. To figure out what to do without her cooperation, he should ask the clerk's office of the local family court.
The court will be reluctant to change the status quo without a really good argument from mom, but he will have to be able to prove that he actually is the father, that his son lives with him (school records, pediatrician and dentist records), and that the mom has been absent (phone records or lack thereof, proof of where she has been living etc.). Even with all of that, it's entirely possible for the judge to award the mother joint or even primary custody and/or visitation. So yes there is a big gamble with going to court, but he really has no choice.
Normally I say that people don't always need attorneys for custody cases (we have full custody of both my son and step-daughter and did it without lawyers) but I really think that in your case, you need to work with an attorney. Is there someone in your families who can help you pay for a lawyer? Mom clearly sounds like a flake so she may be one of those people who will disappear with a strongly worded letter from an attorney. My SD's mother didn't even show up for our custody hearing, but we had warned her that if she fought us, we would throw everything we've got at her (which is a lot, she's a train wreck), would win custody and then would go after her for child support. If she agreed to custody, then we would not pursue child support. Maybe just getting something strongly worded and in writing will make her reconsider her options.