Free & Clear makes shampoo, conditioner, gel, hairspray etc. I have used their gel and Vanicream soap is the only one that doesn't make my skin itch a lot. I like the gel, it's not a very hard hold, but it's a decent gel and it's CG-friendly. Adds shine and definition. Controls frizz.
The shampoo and conditioner have mild (mostly) ingredients and I have read many good reviews. The only bad reviews are for things like detecting the odor of the ingredients (yep, it has NO added scents or cover-up scents!) or that the shampoo isn't high-foaming (yep, it's gentle, you can't use it to get grease off the garage floor).
Here's a link:
You are far worse off than I, but I do empathize and encourage you to check out Allergy Superstore and also search for "unscented shampoos and conditioners" because you will find more than 5 or 10 years ago.
California Baby and Magick Botanicals come to mind too. They're pricier than drugstore brands, but I find you can often use less. often has these products at good prices with reasonable shipping. Free&Clear is available in some drugstores.

As for me, I've given up and begun making my own shampoo, cream wash, and conditioner for me, my spouse, and my dog. All of us have allergies and skin problems. I'm a much happier camper now.

Jessicurl makes unscented products, if I'm not mistaken. That's a popular brand here for curlies.
Good luck to you!