L.,
We're a "try natural first" family as well and also dealt with allergies and skin issues.
You can do a google search for a lab that does blood work for food sensitivities/allergies. A regular allergist will only test for flat out allergies, not sensitivities. You will probably need a naturopathic doctor to help you out, and you may want to look into NAET a safe, natural alternative to manage allergies which I can personally attest to as effective for both myself and my child.
The best thing I found to calm down eczema is emu oil. Jojoba oil or coconut oil are also helpful. Just get 100% organic oils, and make sure you use a natural baby wash, no parabens, no fragrance, no proplyne glycol, and no "hydrolyzed milk protein" (or wheat or soy protein)-which are just "natural" sounding terms for what amounts to MSG. Also watch for sorbitol, malitol, xylitol in body care/food products. Those are artificial sweeteners that also can cause skin issues and allergies - I was shocked to discover sorbitol in many creams, lotions, etc. and when I eliminated all body care products with it I was amazed at the difference - no more skin issues.
The other thing we've had to do is eliminate some laundry products. Nothing scented. No dryer sheets even the unscented ones - just dryer balls. We recently switched to natural cleaning products only - vinegar, baking soda, essential oils, etc. THAT was the final piece for us with all the allergies, including my husband's chronic sinus issues.
It takes time to figure out what is causing the eczema and often it is not just ONE thing, but a combination.
Google gluten free/casein free diets and you will find many ideas of what you can eat. My son had a dairy/casein allergy and I know how difficult it can be.
Congrats on your son! Hang in there, you'll figure it all out. Just take one step at a time, find supportive moms to be by your side, and go easy on yourself. You don't have to be perfect. You just have to love your kid and follow your heart.
L.