The auction committee is interested in using the yoyo quilt for their annual raffle, which is part of the auction fundraiser. Since they take the raffle quilts to various places for temporary display, that may be why they want it backed (due to extra handling and possible hanging).
I appreciate everyone's comments. I think I can tackle putting a back on this yoyo quilt. I volunteer at our university museum, mostly sewing on hanging sleeves, but I have also worked on mending two yoyo quilts. That experience tells me that I would like to make this one as durable as possible.