X-ray producing synchrotrons are generally pretty large and expensive. There has been thoughts of using them for lithography and microfabrication in the past (see LIGA) that haven't really panned out. There are companies working on more advanced laboratory sources, but mostly for analytical applications (EasyXAFS, Sigray). There is a company working on a lab scale synchrotron (lyncean tech).

Do you need to use a synchrotron? Is the radiation generated coherent or something? Otherwise why not use a liniac or cyclotron?

You would really need a free electron laser to produce sufficient photons at the desired wavelength.

The lower the wavelength, the less photons you have at constant power, so shot noise becomes an issue...