This has already been mentioned but I will put the idea in more detail. The whole idea of island planet does not depend upon one, but at least two factors.
Cratons And Tectonic Plates Movement
Cratons are those innermost parts of continents which don't get flooded by seawater no matter how much the sea level rises. On our home planet, earth, primitive cratons appeared long ago. Some ... 4 billion years or so. These were pieces of higher ground which did not get submerged in the water when the primitive oceans formed.
Those primitive smallish pieces of lands were like large chunks of mud sitting on the mantle (the middle layer of earth). These cratons were not stable on their positions, but were rather moving rapidly due to the activity of the mantle those days. Those small islands (cratons) collided and some of them fused into each other permanently. These were the first proto continents.
The underneath part of continents (which sits on the mantle) is known as tectonic plate. Nowadays these plates are slowly moving (their movement was much faster in the past) due to processes known as ocean floor spreading and subduction zones. At places of ocean floor spreading, lava in the mantle spews out on the ocean floor and pushes it off. It is sort of like the earth is trying to get bigger. If there is a region of sea floor spreading between two continents (there is, between Africa and South America), it will push the continents apart.
There is another thing known as a subduction zone. It is a place where stuff from the crust falls back into the mantle. Subdunction zones are present where two tectonic plates are present and one is pushing the other down into the mantle. This is sort of like the planet is getting smaller. The overall effect of ocean floor spreading and crust subduction is that the planet neither gets bigger, nor smaller, but it's tectonic plates move around.
Geology Of An Island Planet
There are two possibilities for you for an island planet.
1- Earth-like tectonic plates with a higher sea level. This means your planet is just like our home planet, only the sea level is higher (the higher the sea level, the smaller and the lesser the number of islands). For this I present you a picture of earth during the cretaceous period (nearly 70 million years ago).
During the Cretaceous, the sea level was much, much higher than it is today. So a lot of what is earth now, used to be part of the ocean then. These regions are marked as light purple on the map which shows that this is part of land, covered by water. These are shallow seas with depth no more than 300 feet.
If you keep increasing the sea level, the purple regions would start increasing and taking up more of the land. This is one strategy for you if you want to have an island planet. The islands would swim about in the oceans (a few inches per year) but they would stay islands.
2- No active mantle activity. If you have a planet where the mantle is not as active as Earth's, then it is possible that the primary cratons never fused together to form larger bits (the first continents). For this, we need an ollld planet (some 5 billion years old). It needs to have an active mantle activity period in the past but now the mantle activity is dead (as earth's mantle would be nearly dead in around 500 million years or so). It used to be a fully terrestrial planet in the past (no oceans, all land) but then a massive storm of water-coments hit the planet and transformed it into a planet like earth: mostly composed of water. Now only the higher grounds of the planet are island and the rest is sea. These islands are not moving even 1mm per year and are stay put where they are.