V.W.
Having worked in a law office specializing in child support enforcement for almost a decade, let me say that your situation is not uncommon. First and foremost, the support is for the child; and visitation (or lack of it) does not alter the support obligation a whit. If the son is visiting you over the summer school break, then you will not LIKELY get a suspension of payments, but it is not impossible, either. The cases I have seen were mostly when the custodial parent agreed to it. Mom, as the custodial parent, must still (whether she is paying out of pocket or not) maintain a residence for the son for when he comes back home. You can petition the court to consider suspension or reduction of support for these weeks/months, even if it is specifically addressed in your husband's existing court order (i.e. divorce decree, or any subsequent modifications thereof), but is by no means guaranteed that you will be granted a suspension.
If the son is MOVING to live with you permanently, then support can be suspended and/or the mother can be required to pay support to the dad. Your post wasn't quite clear on this point (summer visitation vs. change of permanent residence).
Whatever the case may be, if your husband is required to pay support by a court order (i.e., the divorce decree) then he needs to (at a minimum) send a letter to her (copy to the clerk of court with proper references to his case file number) stating the dates he expects to implement the summer visitation/change of residence. If the court has ruled that support is suspended during summer visitation (per the court order already in place) then reiterate that in the letter as well, identifying the dates you will not be providing the $$. If you want the court to consider it, if not already awarded that way, then you need to file a Motion for Modification of the court order, either to Suspend Support (if you will be getting permanent custody/residency) or to Modify/or Temporarily Suspend Support during the months of summer visitation. Any family law attorney should be able to file the proper documents for you. You may even be able to do it yourself, if you will go down to your county clerk's office they may have some forms you can just fill out. But it will require some research, patience, dealing with some getting the run-around (most likely) and basic legwork. If you can afford it, it would be worth the reduced hassle to let an attorney handle that part. Most attorneys offer a free initial consultation. You could use that to assess what your options are, your odds of 'winning' are, and what it might cost to pursue. Additionally, if you are getting permanent custody/residence, then your husband may wish to pursue support from HER. An attorney can help you file a petition for that as well.
But know this: Just as visitation (or lack of) has no bearing on whether the support obligation must be paid... the support being paid (or not) has no bearing on whether or not visitation is enforceable. They are treated as separate issues by the court. So her threats to not send the child (if your husband has been awarded summer visitation through a court order) are empty. If she violates her part/obligations (allowing court ordered visitation) then she can be held in contempt of court. Likewise, if your husband violates his obligations (ordered support) then HE can be held in contempt of court. Or BOTH could happen simultaneously! They are separate issues.
Sounds like your husband needs some very specific terms in the court order about how visitation/support needs to be applied. Sometimes the courts are less specific than others, particularly regarding visitation.
Best wishes.