But, all was not lost! I decided to try out using bias tape, as it would make super pretty edging and save me from having to fold over the vinyl multiple times, which my sewing machine (and I) would not have tolerated.
So, after a few bumps, the project turned out to be pretty painless. I just sewed the two fabrics together (wrong sides together) and used bias tape to cover my pesky raw edges. Here is a really great tutorial I found on bias tape. I also found that paper clips and clothespins worked much better than pins for this project. Binder clips would probably work, too!
The cherry on top: I have just enough umbrella fabric to make two small pillows, thereby turning my picnic blanket into a star-gazing blanket! :)