What material would support the outer ring of a woven children's nest swing?
1" or 1-1/4" flexible polyethylene pipe, such as is used for irrigation systems, would probably do well. It would be rigid enough to hold shape, yet light enough to not knock a kid out in a forehead shot. If you're concerned about rigidity, nest one size inside another.

You should be able to buy it by the foot from big-box home stores, or in small coils at most hardware stores. You'd want to use an interior splicer, or a wooden dowel if one isn't readily available.