A Personal Summons

If you are high level and cannot unlock the portal and cannot find the quest anywhere to start Vashj'r, go to the Hero's Call Board as indicated. If the Vashj'r quest is not one of the three listed, do not despair. Accept one of the other Cata quests, and then the Vashj'r quest should appear on the Call Board screen. If not, keep accepting other zone quests until the Vashj'r quest shows up. Accept it, then off you go!

However, if you have started ANY of the Battle for Lordaeron/BfA starting quests, even the breadcrumb quest in new Dalaran, you will be phased out. I had no uncompleted quests in my log, but I still could not interact with Recruiter Burns to start [Hero's Call: Vashj'r!]. A GM told me that was because I had taken the above breadcrumb so was phased until I completed the whole starting BfA line. I talked to Matthias Shaw on the SW dock, skipped the Lordaeron cinematic, talked to Anduin in the SW throne room, skipped the Jaina cinematic, wound up in Boralus, did the first quest to tag the inn, FP, etc, then was able to go back to Stormwind and do Hero's Call: Vashj'r to open Vashj'r.

Is this quest still available?

This quest is automatically displayed and accepted for you as soon as you enter Stormwind.

EDIT: In later betas this quest was offered by the Hero's Call Board in Stormwind. If you don't find it in your quest log, check the Hero's Call Board.