Receding Chin Repair for Someone Under 18?

Evaluation of the relationship between the upper and lower jaws needs to be established before deciding on treatment. It is possible the the entire jaw is set back too far and would benefit from advancement. This would require orthodontic pre-surgical treatment and a large operation. Clearly this is complicated and expensive.
Formal surgery for a week chin includes a genioplasty (sliding the bone of the chin forward after cutting it) or placement of a chin implant. There are advantages and disadvantages to both. Your plastic surgeon can discuss these with you.
Finally, a temporary fix would be injection of a filler such as Restylane. This would give everyone the benefit of seeing what the augmented chin looks like without committing to something more permanent. In a 15 year old, this is worth of consideration because of the psychologic issues involved.
I very much like video imaging. This is free and gives a good representation of what an operation can do.

Yes, you could wait, but it seems that your son, who is nearly an adult, may need work on his chin at some point anyway, so you could have a chin implant performed now, or wait until later. Summer can be a perfect time for a procedure like this, as he'd have the rest of the season to get used to his new look before school starts again in the fall. Many of my NJ chin implant patients select that time of year. After high school graduation might be a good time, before he enters college, but it sounds as if he's urging you to remedy this problem before then. It's unlikely that, at 15, his chin will "grow" significantly over the next few years before he reaches full physical maturity.

In terms of mouth breathing, there could be several reasons why he’s a mouth breather, such as enlarged adenoids, enlarged tonsils, which would cause this. As a practicing ENT here in New Jersey, I see this often. The next step, in my mind, would be to see an ENT about the mouth breathing and perhaps a facial plastic surgeon as well, so you can understand what's going on before you make any steps towards any kind of surgery. If you can find a physician who is both an ENT and a facial plastic surgeon you can likely get the information you're seeking from one physician rather than seeing two different ones.