We've made good progress developing the components of our project. Here's the inside of the paper towel dispenser unit:We need to attach a motor here, which we're working on a plan for:

In the meantime, we've gotten speech detection software loaded onto our Pi that responds to the command "dispense towel." It uses Picovoice's wake word detection algorithm, and is available at: https://github.com/BrainHu42/spencer-the-dispenser

The paper towels we have readily available (Scott grey core) aren't compatible with the paper towel dispenser unit we were able to get (blue core), so we're going to 3D print an adapter.
We're not sure exactly what dimensions would be ideal. Since the adapter is pretty small and won't use much plastic, we decided to just print a tray of slightly modified adapter designs in hopes we'll get one that fits perfectly.