Where to get a fireplace door retaining clip
Some of the responses got me thinking along the lines of something else I could use in place of the original. 

It turns out that the Hilman clamp fasteners that are in the hardware section at some big box hardware stores are about the right size and shape and with a little bending do the job perfectly.

Thanks to everyone who took the time to help. Your responses got me thinking along the right lines to find something that would work to hold the doors close while still looking OK.