Who holds Dragonstone?
It is unknown as of now according to show canon. Show has not depicted Dragonstone since Season 2 and it is unlikely that they would include it again, except perhaps in Daenerys' scenes. So this is going to speculation. There can be following scenarios:
- Dragonstone could still be held by garrison left behind by Stannis Baratheon. But if that's true, it is entirely possible that the Commander/Castellan might have surrendered to the Iron Throne after hearing about Stannis' demise in wake of 1st Battle of Winterfell.
- Iron Throne may have dispatched a small naval detachment to retake Dragonstone. It is supported by the Tyrell-Lannister alliance as Tyrells had a very strong Navy, mostly because of their bannermen House Redwyne. So the alliance would have provided Iron throne with much needed naval vessels required to invade Dragonstone as Royal Fleet was destroyed in Battle of King's Landing where it was used as bait for luring Stannis' fleet in to be burnt with wildfire.
In any case, it is unlikely that anyone other than Iron Throne holds the Island.
However, one thing to note is, in Battle of Castle Black, Stannis had ~2800 men. Four episodes before that, in "The Laws of Gods and Men", he says that he has about 4,000 men left overall. If he appears on the wall with ~2800 men, that means he may have left 1200 men as garrison in Dragonstone. Alternative explanations can be that those 1200 men were left behind at East Watch by the Sea.
Importance of Dragonstone
You are incorrect when you say that Dragonstone is the seat of naval defense of King's Landing. That is simply not true.
Dragonstone is a damp and bleak island with very limited resources. It has a port and traditionally hosts few squadrons of Royal Navy but King's Landing on the other hand is a bigger port and main power of the Royal Navy resides there. However, it lies very close to the Gullet and can be used as a base to impose naval blockade on King's landing if the besieging forces have enough naval vessels.
In any case, there is not much strategic importance of the Island.
But of course it has sentimental value for Daenerys because:
- It is the ancestral seat of House Targaryen. Since Doom of Valyria, All the Targaryen Lords, Kings and Princes (From Aenar the Exile to Prince Rhaegar) have held Dragonstone as their seat1.
- Daenerys was born there.
- Daenerys' mother died there giving birth to her so it is plausible that her ashes might be interned there (Since end of Targaryens were writ in flames unlike common men who were buried).
Can Daenerys take it?
Dragonstone is a formidable castle but like every other castle, it is vulnerable from the air. Daenerys has three dragons and therefore aerial superiority.
She also has a strong fleet to besiege the Island from the sea and prevent any reinforcements coming from the mainland. It is unlikely that a large defending force would be stationed in Dragonstone.
But If I were her, I would rather land in Dorne because:
- Dorne is much closer when travelling from Meereen as compared to Dragonstone. There is no point in braving the danger of sea voyage unnecessarily.
- Dorne is allied with Daenerys along with the Reach. If she lands there, she will find her Westerosi allies and their armies waiting for her, making the invasion all the more easier. If she lands on Dragonstone however, she will probably find foes and waste time on siege and extra travel to Dragonstone. Then she will need to wait for her Allies in Dorne and Highgarden to assemble and March Northwards to King's Landing.
- Dragonstone holds no strategic value for her.
In the Books:
Dragonstone was occupied by Iron throne with a force under command of
Ser Loras Tyrell after Stannis departed for the Wall.
1. Targaryen Kings held Dragonstone as their seat only during the period of time they were nominated Prince of Dragonstone by the incumbent King. Although many Kings did not hold either the title or the Island before their coronation. Princess Rhea, Viserys I, Prince Joffrey Velaryon, Aegon III, did not hold the land despite being named Prince(ss) of Dragonstone. Baelor I, Viserys II, Aerys I, Maekar I and Aegon V held neither the title nor the land prior to their coronation.