Thanks for the advice everyone! Since I want to cover such a big area, the more I think about it, it will be a lot of work to install so many grommets. I have a lot of scrap wood laying around that we use to burn in our wood stove. So I'm thinking about either using wire ties or zip ties to attach the wood to the fence and staple the plastic to the wood.
Sounds like the best idea for attaching it so far. Only thing I would maybe do differently would be to add a second board covering where the staples are. Extra reinforcement to help prevent tear out.