I'm not sure, but I personally wouldn't bank on that. If you don't complete online check in before it closes,
DCL might assume you're not sailing and cancel your reservation (kind of like how airlines will cancel your ticket if you don't check in XX time ahead of your flight). I know they can manage/update documents at port if something went awry during online check in, so theoretically you could show up and just do it all there. But I'm not sure that they'll allow it.
If you're not angling for an early PAT (which, if you were willing to just show up at port with your documents tells me you're not), then you really have nothing to worry about. Many of the tips here are how to get through the process quickly to get an early PAT. And even that has spaced out since they tiered online check-in by
Castaway Club status.
Just have a photo of passports or birth certificates for each person in your party and 'headshot' photos for each person in your party (looking straight on, shoulders up, no hats and no glasses if they cause a glare). You'll have to hand-enter information for each person-their DOB, address, passport info (if it doesn't auto-populate after you upload the picture), emergency contact, travel plans to/from port, and how you want to handle your onboard account. If you're not rushed for time, just do it anytime before online check in closes, at your leisure, and you'll be fine.