Oooh, found out a neat trick!
- Right click a screenshot
- Copy its url, take out only the file name, (no path, no file extension)
- Go to archive.org, select the Wayback Machine bar
- Type "miiverse.nintendo.net/posts/", and paste the file name.

My Miiverse data came in the digital mail yesterday. Check your inboxes!
EDIT: the links expire 12/31, move fast.