As this is a surfboard building site , there is a video of a Japanes kid making a cardboard surfboard in his living room , its on the wavescape surfsite and there is a link to a company making the boards www.sheldrake.net.
It has been done many times, going back to the 60s. There was even a guy who posted here that made them from pizza boxes. He didn’t surf, and had no clue.
Looks like lamination is a very long and tedious process.
This guy Sheldrake used to post on Swaylocks. He’s shown the process and did ride reports. Try searching for it.
“Keep an eye out for BIG changes in the boat industry” - ad from 1960. 
It’s actually about the same weight as wood. Makes sense if you can get the cardboard for cheap for free… otherwise, not so much. Heatshrink over the cardboard before lamination maybe…?
Take on a little water and the cardboard core is mush.
But exaggerated, because its cardboard. You could try sealing the cardboard, but then you increase the weight.
I built and rode several hollow wood boards, but found them impractical because they would take in water from the smallest hole or ding, then drying and repairing became a big issue.
They are novelty boards, not really practical as a daily driver, or less practical than a foam board anyway, but that alone isn’t a reason to totally dismiss them. Novelty boards can have their place too, and can be made more viable with vents and good glassing.
The Sheldrake board seems like it begs a wet-out-table type of glass job, but surprisingly its never mentioned in the context of glassing the cardboard frame.
http://s238.photobucket.com/user/fishersfort/media/fr_1057_size880.jpg.html
With a wet out table the glass is squeegeed with resin on a flat table, rolled up, and unrolled on the surfboard. The problem with trying to glass over an open framework is that there is no surface resistance below the glass to push against while applying the resin. I’ve never used a wet out table, but after seeing how they work, I think it would be the way to go for an open framework like the Sheldrake.
As shown in the video, it is hand lamination with a brush:
The following link is where the lamination portion of video begins:
https://youtu.be/enfZlfjFU4c?t=252
Full lamination video: