A contract manufacturer does not have to register and does not list.
The company engaging you, presumably the eventual marketer of the device, must be registered...usually as a Manufacturer or a Specification Developer...and must list the device.
Most Class II devices require a 510(k). This would be obtained by the company engaging you.
As a contract manufacturer, you could not obtain a 510(k) directly. Your customer however could include in their list of services they hire you to perform, provision of support-services for their 510(k) effort such as design control and technical writing.
