Otakon used to provide the plastic with the old paper badges. The new, more durable ones debuted last year, and there were no plastic holders given then either. So to my knowledge answer to your question is neither
Actually they did have a few, although I don't know if I was just lucky or not. The little plastic part of my badge broke last year due to someone stepping on it ( it was my own fault for setting it outside my bag during a photoshoot ) and they were more than happy to provide me with one. it didn't fit perfectly, but it got the job down. I do second the clear tape, it does wor