Hi. I have severe chemical sensitivities, so somewhat similar though not as critical as your son's situation. But as a mother, I can understand how much you want to do this the right way for his sake.
I would like to approach the scenario from another direction, however. First, obviously, stop using any chemicals on your own lawn and yard, maintain a buffer zone between your neighbors so he is within the "safe" area. In and out of the house, eliminate/substitute every suspect product, including bath and laundry, personal care products. In the kitchen, I didn't realize for example, that Ivory Liquid dish soap could cause asthma. I was hand washing more while trying to look for a "safe" dishwasher detergent....
Now, regarding the soil mix for your raised bed, realize that if your son won't be playing "in the dirt" he will be one step removed from direct contact with whatever is in the soil. You may want to look at it in MORE THAN SIMPLY ORGANIC APPROACH -- meaning rather than the soil less vermiculite, peat, etc. which are not "living soil food web". There is an excellent thread about this that I will find the link for later....
Found it: Subject: Organic? or sorta
When the soil is alive with diversity of microbes the pollutants will undergo several iterations of biological processing as they are digested and broken down, so that by the time they are chemically recombined to be used by the plants, they may not be the same as they were before. a GOOD compost will contain the biologically diverse beneficial organisms you want to introduce to your soil. They are the equivalent of good yogurt with multiple live and active beneficial bacteria, good yeast bread, lacto-fermented foods, and all the enzymes.
That's how I view my garden.
You do need to understand that at the same time, there will be more exposure to a wide variety of microbes. Until your son's sensitivities are well understood, and his immune system is strengthened, there may be some that he will react to. I happen to have increased allergic symptoms during the worst pollen and mold (fungal spore) seasons. Obviously wash hands well after coming inside, maybe even change clothes and at least wash your hair, and protect him from exposure to these allergens. I have an app I use as a guide called Weatherbug which displays allergen levels in the area.
Personally, IN ADDITION TO MY OWN HOME MADE COMPOST, I like using well composted organic bagged soil and soil amendments -- the ones I have been using are Bumpercrop garden soil amendment and their potting mix Gardener's Gold, Espoma products, and Dr. Earth products. But if you live on the West Coast or In the Mid West, there are a couple more brands that seem to be also highly recommended but I'm not familiar with.
BTW peat moss and vermiculite and perlite are products I no longer use (though some of those organic potting mix still contain peat moss) because they are usually sold dry as dust and the dust particles from any of the three have cause severe problems for me when inhaled. Good organic potting mix and soil amendments containing compost are bagged and kept in a wrung dry moist state because the microbes will die off if completely dried. And good gardening supply distributors will keep these bags in protected areas out of sun and rain. Don't buy waterlogged bags,
I don't know if that answered your question. I hope so.