I just realized that the HoF skipped work to go look for the cure.

The First Warden is more a figurehead than a real leader. For all intents and purposes a given Warden-Commander has no superior.

I like to think that my Wardens used some vacation days or forged a doctor's note claiming seasonal allergies kept them from visiting Weisshaupt. I could see some of my Wardens sending off an FYI birdie. But some others have probably forgotten the dude exists.

I'm not sure they necessarily needed the permission of the First Warden provided they made sure someone was covering their absence. Vigil's Keep was an important experiment for the Wardens and the First Warden did seem to be keeping a close eye on it at the beginning as a result. However, after roughly 9 years with hardly any real problems, I dare say they were content to trust the Hero to know best how to run things.

What was harder to understand was why Clarel was interfering with the management of the Ferelden Wardens. In Last Flight this appeared not to be the case, since reports were coming from Vigil's Keep to Weisshaupt as per normal and it was only Clarel that had fallen silent on them. Even this did not unduly raise alarm with HQ as they were accustomed to the local leadership around Thedas occasionally failing to report. They could, after all, be away on a prolonged foray against the darkspawn.

This is where the novel and the game differ in how matters proceeded in the south since it implied in DAI that all the Wardens across Ferelden and Orlais were affected by the false calling and that Clarel had assumed leadership of them all. So, perhaps we can assume that on route westwards, the Hero (or Orlesian Warden if the Hero is dead) stopped off to speak with Clarel and asked her to keep an eye on things whilst they were gone. That would explain why she was seeking permission to enter Ferelden from Teagan because this wasn't an official arrangement with the approval of the monarch prior to the Hero leaving but only a casual arrangement the Hero made with her as they were passing through.