Vartag can be found in the Chamber of the Assembly in the Diamond Quarter. He will offer a task to enable the Warden to prove their loyalty. After completing it, he'll bring the Warden to see Prince Bhelen.
If the Warden is of the Dwarf Noble Origin, Vartag will be suspicious about offering support to Bhelen and it is required to be persuaded (Improved Coercion is needed) in order to believe that the Warden's intentions are genuine. If the Warden is of Dwarf Commoner Origin, their half-sister Rica Brosca can introduce them to Vartag.
If you choose Lord Harrowmont over Bhelen, he remains at the palace after the coronation and will attack you the next time when you talk to him, accompanied by some guards.If Bhelen is chosen to be the king and the Warden makes the ultimate sacrifice in the end, Vartag appears at the Warden's funeral, presumably as the official representative of the King of Orzammar. Alternatively, if the Warden still lives, he attends the post-coronation ceremony in Ferelden's Royal Palace.
You can steal a Champion's Shield from Vartag, up to three times. Once when you first talk to him to get the quest to help Prince Bhelen at the Chamber Assembly, the second can be obtained if you complete the proving fights causing him to show up at the Proving Arena, and the third time after you've finished the quest and in the Royal Palace.