Yes, anyone can ship to a FFL, but some FFLs have policies of only receiving firearms from other FFLs.
You need to check to see what the FFL wants. Some FFLs require a copy of your ID, which is not required by law, but then again they are also not required by law to accept it from a non-FFL.
There is a form you can fill out:
but it has been said that the records are not really removed.