Its difficult to buy bedding without plastic packaging. Search for organic linen and hopefully you’ll find a manufacturer who has considered the packaging. For example, organic cotton sheets from Fern Bird come in cellophane which breaks down in the compost. 
Amy from Auckland emailed saying; In Nature manufacture organic bedding including pillows, duvet inners, sheets and mattresses, right here in Auckland. I bought a few pillows and a duvet inner and it all came in recycled cardboard packaging with no plastic.”