There is no single guidance that is comparable to 3.9 B 16. US FDA's contract manufacturing rules are fragmented throughout their regulatory system. It's fairly straightforward to discover them, though, by going to the medical devices section of the FDA website
http://www.fda.gov/MedicalDevices/default.htm and entering into the search box "contract manufacturer".
It's also useful to search on "specification provider", which is the US FDA term most closely analogous to "private label manufacturer".
It's important to recognize that US FDA historically regarded most contract manufacturing as mostly closely analogous to 3.9 B 16's section 2 scenario C, with most contract manufacturers (if they did not distribute the product on behalf of the specification provider, and excluding contract sterilization providers) not subject to most regulations since they were to be supervised by the specification provider, who was legally responsible for their operations. This has changed somewhat in the regulatory-year-2013 system, with all contract manufacturers now required to Register and List the devices they make, and who they make them for. It still is the case, though, that US FDA places most regulatory responsibility on the specification provider, not the contract manufacturer.
Is there a way to register products as a PLM/OBL and which regulations do I have to fullfill regarding the quality management system (21CFR820?)?
As a specification provider, you have primary and full responsibility for the medical devices you have contract manufactured. You must Register and List, and you must have a fully compliant QMS, particularly including post market surveillance/reporting. Note that significant compliance requirements are found in other sections of 21CFR, not just in 820, and that it's necessary to take guidances into account since in many respects they override the apparent language found in 21CFR.