Categories on a Page

I have looked high and low, used quotes and -posts on Google, and I keep finding widget plugins and theme hacks. (tl;dr – I have searched seemingly endlessly before coming here for community support)

What I am looking for is something similar to this page:
https://magazine.pomona.edu/archive/

I would like to display each category (with the ability to chose selected categories – four each year), with a thumbnail for each category. I don’t want to show posts or anything like that, just a thumbnail with the category link underneath.

Is there a plugin available that does this? I’ve found a few that seem to come close but require core changes and/or are outdated with no support.