Here is my attempt at the cheapest CNC foam cutter that can be built for the home shop. It was inspired by Alted's Low Rider and djk4linux's needle cutter.
Assembly instructions and a parts list is at https://www.flitetest.com/articles/another-cnc-foam-board-cutter.
My plates were milled from 3/4" HDPE that I found, but plywood should work. My wheels are 69 mm diameter from an old pair of rollerblades. The plate needs to be adjusted for different size wheels. X_Carriage_Plate.dxf is for 69mm wheels, and X_Carriage_Plate1.dxf is for 60mm wheels.
The z-axis rails are 5/16" x 8" bolts or 8 mm rods. The bearings are 608zz. The Y-axis tubes are 3/4" conduit.
The slots in the plates allow for a table between 3/4" and 4" thick. I am using a 32" door and get 24" x 69" of cut area. The tubes can be adjusted from a single sheet of foam for a waste board up to a 30 mm thick vacuum pad.
More details to follow...
This cutter was developed from ideas on the Flite Test Forum http://forum.flitetest.com/showthread.php?24251-Cutting-foam-sheets-with-a-needle!
Update: Fixed the location of the bearing pocket in the Y carriage.
Update: Added alternative smaller Z-Carriage it is threaded for a 5/16" threaded rod (or cut off carriage bolt). The original uses a t-nut.
Update: Added Flywheel from "MPCNC 2826/2822 Brushless RC Motor Foam Cutter by dkj4linux" and the Pineapple Coupler from "Mostly Printed CNC 525 'C-23.5mm OD' by Allted."
Update 10/2/2017: Prototype is working! Added vacuum nozzle. Added updated X carriage plate drawing that should work with 60 mm wheels from Vicious1.com.
Update 10/11/2017: Added some thickness to the lower printed bearing on the Z-carriage.