After consulting my cousin, I decided I'd try drying the peas for storage. This has turned out to be the absolute easiest way to store them.
I picked the pea pods, a few each day as they were ready, and shelled them. I put the fresh peas in a small dish open to the air. I left them there for a week or so until the peas were small and hard.
Then I put them in a jar and labeled it. I did add an oxygen absorber in the jar.