Thanks to Pinterest, I got the idea I needed for a solution. I started with buying a cheap ($6) soft wood frame from Hobby Lobby. Took several of the ribbons that I either don't care about or are too short to really be tied around my kids heads, and stapled them to the from (using my upholstery staple gun).
That gave me something to do with all those clips:
Then I screwed these hooks into the bottom of the frame (this I did by hand, since the wood was pretty soft)
And that gave me something to do with the headbands. VOILA!