Mike,
Agreed on everything that Mark wrote--
A little more info on CAD/CAM...Design work can be done in any software that you're comfortable with. For my designs I use a software called GeoMagic, which is a much less expensive version of SolidWorks. When I'm done with my design, I export either 2D drawings or 3D stl files. However--at some point you will need to create toolpaths to cut out these models. The combination of Fusion and vCarve should be able to serve your needs--but vCarve is exclusively a windows program--you could let your students do their design work on chromebooks and have a couple of stations for vCarve that they move to once they've finished their designs...toolpathing is the faster part of the CAD/CAM process so that might work just fine.
Looking at the design in the videos, I would allow at least an hour per car. Flat shapes are very quick and simple to cut, just a few minutes...but complex 3D shapes require the use of a ball nose bit that makes a number of passes to sculpt the shape. I'm assuming that your students come up with all kinds of cool swoopy designs for their cars that would need to be carve in full 3D--which would likely require an hour each.
The handibot costs 2895--shipping averages around $60--and I would recommend a handful of ball nose bits that you might spend $500 on over the next couple of years. You're also fine using the less expensive bits available at the hardware store if you're just cutting soft pine.
Brian