If I was to fix this, I'd do it much like with fiberglass, but with wood glue, sawdust, heavy paper, and solid wood.
I'd mix Titebond III and coarse sawdust to make a paste to fill in the big holes. The outside vinyl should hold it, but I'd make sure that doesn't bulge or get lumpy. Then, I'd glue down a layer of heavy paper (grocery sack) soaked in glue to hold the sawdust fill in place. You could also use thin cotton fabric (muslin) soaked in glue. After that, I'd glue down some thin strips of solid wood, like poplar. The sawdust/glue mix isn't all that strong, so I'd back it up with solid wood.
Titebond III is thinner than II and will work better for soaking into paper or fabric, but II may work better to make the sawdust/glue paste. You'll have to play around a little and see.
My repair linked above is holding great!