I was able to resolve the issue.
I had wireguard tunnels open to another organization. I cycled through and removed all tunnels with fly wg remove <org> <tunnel>. After removing all of the tunnels, I re-ran fly doctor -a <app> and the command just hung. I went into Activity Monitor and force-killed the Fly Agent. I then started the fly agent again with fly agent start.
After cleaning up all the tunnels and rebooting the fly agent, I was able to run fly doctor -a <app> successfully. Then everything worked again.
EDIT: This happened again this evening, except there were no wireguard tunnels set up at all (see removal above). Turns out the Fly Agent was hanging. I had to force quit the fly agent via Activity Monitor, restart it, and things are fine now.