Try Green Building Supply wool carpet. I haven't tried their carpeting but have purchased other items from them. It's a good company.

https://www.greenbuildingsupply.com/All-Products/Wool-Carpet

Yes, Green Building Supply is a smart bet for all flooring, including carpet!

Earth Weave is among the best non-toxic choices. But the carpet itself is just one part of the answer. You've got to pay a LOT of attention to the padding underneath, too. I wrote about a lot of helpful tips on how to find the best allergy friendly flooring at: https://www.getgreenbewell.com/essential-guide-to-allergy-friendly-flooring/

I think it's interesting that carpeting can sometimes be the BETTER choice compared to hardwood flooring. A lot of people assume carpet is bad, but it might work better for reducing chemical exposures than some low quality hardwood flooring or vinyl flooring.

Only floor coverings made of natural fibers come to mind: wool, cotton, jute, hemp. The challenge is finding wall-to-wall coverage in these.