Hi @Metamer, thanks for checking in! Yes, other webcams should work (we've been using webcams that are designed for laptops because the form factor works for our setup, but other cameras should work as long as you're willing to tinker with the placement somewhat).
As an alternative, we're currently selling lego spectrometers that work with the Raspberry Pi camera (more about that version here) at our online store. If you look at the pics/instructions for the lego version you should be able to make it work with your desktop kit without too much fuss.
We're working on sourcing replacements for the webcams that are in the Desktop Spectrometer, so they will be back in the store soon, but at this point we can't commit to a precise ETA.
Let us know if this helps! If you drop us a line at email@example.com we can let you know when the webcams are back in stock.