What is the minimun age to work in a machine shop.

Depends on the job. You can get qorking papers at 14 but those limit what you can do.

At 18 you can work anywhere that will hire you.

You are able to work at 15 as long as you get working papers from your school but if your grades go down they can report it to your work and they will pull your working papers

The best thing to do would be check with the machine shop. You may be old enough to get a work permit but if they can't hire you because of restrictions, that work permit won't do you any good there.