As you begin wedding planning, start gathering & completing these documents ahead of time, as they must be submitted at least 4 weeks before your wedding date.
Marriage Notification (Hjónavígsluskýrsla)
Marriage Application (Announcement of Marriage)
Certified birth certificates
Certificate of marital status (Issued from your home country and can't be issued more than 8 weeks from your date of marriage.)
If you are not in Iceland, you can mail the documents, or even reach out to the Registers and see if you can scan everything in, but keep in mind need to turn in the originals no less than 5 days before your wedding if you go this route.