How old is the tree? Either it’s not so old or your neighbor is extremely lucky. The curculio and brown rot are nearly everywhere on the east coast, and with either of those strongly present in your orchard and no spraying you will get no peaches at all. Apples are another matter, I’ve had plenty of no-spray gnarled worm-infested apples that tasted great. But the topic here is peaches.
Back on the original question, it should not be impossible to grow some OK peaches with only organic sprays. Surround will make a big dent in the curculios and moths, and copper and sulfur will let you get a decent portion of rot-free fruits. I went that route for about ten years but got tired of peaches rotting right after they were put on the counter. Now the only non-organic thing I use is Indar / Elevate for brown rot control. My most rot-resistant peach is Indian Cling, it did a bit better than any other peach I tried (about fifty varieties).
A full guide of my methods are posted here, and I include a fully organic option. You don’t need to do all the things here, it depends how good a result you need.