In light of the fact that I make cams for a living, I can speak with a high degree of authority on this particular subject.
Depending on how much you want to spend and how much programming exeperience/expertise you have access to will determine how or if you can construct such a device. It is NOT easy to do (synthesize the cam lift data PROPERLY). I can say from years of experience, a CMM is categorically useless for measuring cam lift and/or investigating profile dynamics.
Frankly, we offer the service (reading cams) to the hobbyist and/or performance engine builder. We do it for a small fraction of what it would cost to build a machine and a smaller fraction yet for what it could cost you to develop one.
The same device we'd use to measure your cam is used to measure the cams we make for OEM prototyping and/or our race cam clients.
We can graph out and supply the full dynamics (lift, velocity and acceleration plots) or else simply supply a simple cam lift chart. It all depends on what you want/need in the way of data and the amount of data you can deal with/use.
We also offer cam grinding as well as profile design services. Total valvetrain engineering services are also available.