@JeneLitsch Welcome to the Tiled forum!
Thanks for bringing up this issue. It is because snaps are sandboxed and Tiled only requests read/write access to your HOME directory. Apparently external USB drives need a separate permission. I’ve added this now, so if you install the Tiled snap from the “edge” channel you should be able to access files on USB drives:
But of course you can also keep using the itch.io version.