I signed into the FRCTools page and left it open for a couple of weeks. When I came back to it, there was a red banner displayed that said "Unable to Load Orders - No organization selected" even though I could see my organization name displayed (selected) at the top of the window. The real issue appears to be that my auth token expired, not that I had no organization selected. Once I signed out and signed back in the banner was gone and the app worked as expected.
Ideally, the behavior when a user's auth expires should be explicit that the user needs to sign in again. Just displaying symptoms of the user not being signed in (like the red banner I was shown) isn't very user-friendly.