Tampons are made from cotton, not plastic. Maybe you're talking about the applicators (which are also sometimes made of cardboard instead of plastic)? Or maybe you're talking about menstrual pads, which - like diapers - have a plastic lining to protect against leaks.
Is cotton production significantly affected by oil shortages? I thought cotton was an issue mostly because farming it is water-intensive and too much of it is produced in dry regions.