Can someone use the Woodshop drill press for shaping resin?
What about the sanding wheel in the metal shop?
They also are wet sanding using the drill press with a towel under it to protect the press from water.
Or would this better be done on the lathe?
Use the best tool for the job that you have available. Luckily we all have access to a plethora of tools soโฆuse the lathe 
I have been cutting a lot of carbon fiber on my drill press at home with a diamond hole saw and water. I just wet vac the sludge and wipe it down with paper towels and oil. I have heard of people using rubbing alcohol in place of water though.
The best result for shine is one last layer of liquid plastic but Iโm about to lathe a wood/resin project very soon.
Resin on our sanding disk would gum it up, and itโs not readily replaced by a user. I would say donโt use it for stuff like that