Filemaker could create a new record and insert an existing PDF into a container field in that new record, but "print a new document" means what exactly?
By print a new document, I'm thinking that from an open document, 'Cmd P' would present choices, say, 1) HP LaserJet, and 2) Filemaker. Picking #2 would open a target database and new record, with the 'printed' document in the live container; the same functionality would ideally also exist in the scanner, where a newly scanned document would be sent directly to this 'printer' and a live container in a new record.
This is the way my PC application works - it was very convenient. Now, I print PDFs into a holding folder, open FM, and drag and drop the PDFs into the live container in a new record. I'd like to eliminate that step - if possible.
FileMaker can be setup to insert all PDF files found in a particular folder, but the "cmd P" part of what you describe requires changing the behavior of parts of your system external to FileMaker and not being a Mac User I don't know if that is even possible.