Webdav: how to identify outdated custom modules
I suppose it might just be a matter of checking whether anything is present in the associated folder. Each subfolder should have an .info.yml file that contains the version of the module, which in turn can be compared to drupal.org/project/<name_of_module>/releases.
Evaluate best option: script/tool/job.
Eg a cronjob runs a program which writes to a basic FIFO queue.(site admins/ module versions, links). Connect a queue consumer that would call the post /notifications api to reach users about outdated modules and available releases.
Further: could replicate the work the bot does in MR's to a clone. However I would not move in this lines - would prefer to phase out webdav gradually and offer only composer.. a user which does not handles composer and MR/updates of versions cannot handle custom modules.