A few ideas to flesh out the club:
1. A "punishment tray" - Since removing of panties is so common, they could have a special tray where the spankees put their panties, as well as jewelry if necessary (necklaces etc.)
2. Some hand-made implements - There are basically two ways to obtain an implement for the club: a member brings it, or a member makes it from the scratch. The manually dexterous members could make some paddles since those are probably the easiest. I've uploaded my latest story (Right Tool for the Job) which is not a school story, but illustrates some nontraditional approaches to paddles.
3. Essays -- as this is a research club, some spankings might come with the requirement (for both spanker and spankee) to write an essay about the experience afterwards. A picture of a well-spanked girl sitting on her bare bottom and writing the essay (handwriting, for extra punishment) would be fun
4. A shack -- a room in the main school building would attract too much unwanted attention, so I think that the best place for the club is a shack on the school grounds or a room in an old, unused building.
5. Guests -- a parent might occasionally appear in the club in order to tell the members about punishment in his/her youth -- as well as the ones he/she gives today. Usually ends with a red bottom for the parent's child who is used as an example, as well as several others who wish to try it.