Amazon usually requires products sold on its marketplace to have GTINs, such as UPCs. In the past, being enrolled in Brand Registry provided an exemption from this for sellers of products without UPCs.
Now, you can apply for a UPC exemption with or without Brand Registry here (Seller Central login required). You will need a letter from the brand owner confirming that the products do not have a GTIN as well as a link to the product website, or images of the products.