I would think Vader's main job for most of the time in between RotS and ANH was hunting down and killing remaining Jedi. This would also include training Inquisitors, Palpatine certainly wouldn't have bothered with that. He would have existed outside the standard Imperial command structure, but also able to override most of the officers in whatever situation Palpatine would have sent him to look at. The exception to this would be those officers those at a certain level like the Moffs and maybe Grand Admirals, Vader would be under their command.
Remember Anakin was pretty high ranked in the Republic Army, so he would have been familiar and comfortable with military protocol.
After ANH you have an emergent Rebel threat, with the bonus of the person who destroyed the Death Star rumored to maybe be a Jedi, which would have caused a good chunk of the military to be reorganized as Vader's personal hunter squad. This is even ignoring the fact that he knows Luke is his son and would have gone straight after him no matter what. Palpatine just gave him more resources to be able to do the job properly and make sure Luke got captured so he could eventually replace Vader.
Edited by Monocle, 23 July 2019 - 08:08 AM.