In Rogue One, we see the Director meeting with Vader on a volcano planet. Is there any canon that says were his base is? It doesn't quite look like Mustafar.
if it's the one I think it is, it's the one from Ep III.
After the end of the Clone Wars, the Galactic Empire took control of Mustafar and maintained secret facilities on the planet where Jedi survivors of Order 66 were interrogated and executed. Rumor had it that Darth Vader carried out these interrogations, torturing the Jedi for any useful information before their execution. Additionally, both Vader and Sidious often returned to the planet, using it as a focus for meditation on the power of the dark side of the Force. Four years before the Battle of Yavin, an Imperial fleet was stationed above Mustafar commanded by Grand Moff Wilhuff Tarkin aboard his Imperial I-class Star Destroyer the Sovereign. During a mission to rescue Kanan Jarrus, the Sovereign was destroyed during a duel between Jarrus and the Grand Inquisitor when his lightsaber was cut in two by Jarrus and fell into the ship's power core, sending the vessel falling towards the lava fields below
In 0 BBY, Darth Vader had a personal abode on the very planet where he had been disfigured. Shortly after an insurgency on the desert moon of Jedha, Vader was in a bacta tank, out of his armor, when his servant Vaneé admitted Director Orson Krennic, who wanted to discuss the workings of the top secret Death Star.
As per film novelization, it's Mustafar:
His shuttle pitched and rolled as it entered the atmosphere, riding rivers of black clouds tossed by howling winds. The shuttle’s stabilizers and gravity units maintained an interior equilibrium, but Krennic found the experience no less unsettling for the relative lack of discomfort. He grasped his chair tight, kept his chin raised, and reviewed for the sixth time everything he knew about Darth Vader, recounted every one of a dozen tactics he might use in conference with the man who had made Mustafar his base of operations. (Chapter 13)
The film's junior novelisation indicates that Vader's fortress is on Mustafar, the planet where he and Obi-Wan fought their iconic battle at the end of Star Wars: Revenge of the Sith.
When Krennic arrived on Mustafar, he’d ordered his pilot to land next to Vader’s monolith, the tall black tower the Sith Lord called home. He’d left his death troopers inside the shuttle and entered the forbidding place alone.
As to why Vader would choose to live here, that's answered by the Rogue One: The Ultimate Visual Guide factbook. In short, the Emperor personally commanded him to base himself there.
Vader's personal abode is an obsidian tower on an inhospitable world. It is by the Emperor's design that Vader lives in such an unforgiving environment ... The stark, modern structure is built over an ancient castle full of dark secrets.
We can only surmise what the Emperor's motives were, but it's certainly in keeping with Palpatine's twisted sense of humour to make Vader live in the place where he suffered his greatest defeat.