I had my wood pieces laser cut. There are services online that will cut your shapes and send them to you. I have access to one for personal use so I unfortunately cannot speak to which service will work best for a project like this.
One thing I cannot stress enough with a project like this is make a test sample out of something light weight and easy to cut like chip board so you know if there are layout or sizing errors. Even if you use a laser cutting service, you can print out a cardstock/paper model on your home printer for a test run.