Is there a no sew option for making a papasan chair cushion?

This is one way: https://inmyownstyle.com/2010/07/how-to-make-a-no-sew-cushion-cover.html

Cut 2 pieces of material 6 inches wider than the size of your cushion. Use sharp scissors and cut 3 inch wide strips all around. Type, no-sew blankets, into the pinterest search box to find many ideas.

Velcro to the rescue. Just get a cushion and make a pattern to cover and instead of sewing, use Aleen's glue products to glue the velcro on. Easy to remove when it needs washing.

Sometimes on Craigslist you can buy a just cushion and/or chair really inexpensive

What I'd do is head out to a store that regularly has these and measure the diameter of the cushion. Or measure the diameter of your chair and add 1-3 inches to accommodate the depth of your cushion plus at least an inch for the

seam allowance. Then cut two round pieces of fabric.

But that's a problem since most fabric isn't going to be wide enough to accommodate the width of the cushion. So you'll have to put two pieces of fabric together using either fusible web which is cut to size and ironed to fuse the pieces together, or you can sew (I know I said a bad word) or you can use fabric glue.

Since there is no cushion, you can't Frankenstein the old one, so you'll need to get either a cushion or batting to fill in the new cushion. Batting makes more sense to me having sat in one of those chairs. A cushion might make the chair more like a trampoline than something useful. The batting can be pieced together without sewing or gluing. Or pillows: https://www.wikihow.com/Make-a-Papasan-Chair-Cushion

How to keep everything together? I'd use fusible web which you'll find at fabric stores. It's ironed in between the fabric to fuse it together . You can use fabric glue, too.

You can also tuft the cushion as in the pillow example which is sewing but fairly easy. Traditionally, the cushion is tufted to make it behave better in the chair.

Fabric glue. I've used it and it works.

You know, you could find an old quilt or three and use that as the batting. Goodwill or Savers to the rescue.

BTW, can the child who you are sewing for help lift the machine?