You can access it via the JS Console (ctrl+k in firefox, or right-click somewhere on the page, click on something like "inspect element" and then choose the "Console" tab).
DISCLAIMER Don't randomly copy-paste things into your browser's JS Console when you don't understand them. Neither here nor on any other website. The console is able to read your browser cookies and other secret information and can send it to anyone. If someone tells you to run something from the console you should read all of it and make sure you understand what happens.
Go to any page on wallhaven.cc and type the following in the JS Console (confirm with the Return key):
document.body.innerHTML = localStorage["wallhaven.seen-wallpapers"]
This will replace the page content with the list of seen wallpaper IDs from your localStorage. You can copy-paste that anywhere you like, save it in a text file.
Later you can restore it from the console again:
localStorage["wallhaven.seen-wallpapers"] = "[1,2,3]"
Where
[1,2,3]
is what you previously copy-pasted. This will most likely be a lot longer. Don't forget the "quotation marks".