First of all thanks in advance. I need suggestions on how a five man special ops team can sink an island entirely.
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If the island is small enough and the SoF team has access to modern mechanized transport, they can always transport themselves and a thermonuclear device on location and find a way to place the device below ground (either a natural cave or cavern, or an underground bunker, service shaft or similar construction).
Of course these guys need to have been working out for a while, since to entirely remove an island from the face of the Earth needs something like "Castle Bravo"
Castle Bravo detonation
OK, so maybe it isn't technically "sinking", but the crater will be well below sea level when finished.
Either the island moves down or the sea level rises, both have their difficulties (understatement) and side effects.
Islands are attached to the Earth
So first off we need to consider that to move an island down you need some way to move enough of the material that lies between it and the sea bed, they don't float so can't be 'sunk' as such. This all changes if you've designed a floating island but I expect you would have mentioned that.
The easiest way would be for the (small) island to be sitting on top of a cave system full of air or underground lakes - possible if your island is sitting on a layer of softer rock. If your island is on a fault line and your five man team is willing to put in a lot of time then you could cause an earthquake to bring the island down to fill the gaps the air/water once occupied and leaving the space above to be filled in by water.
Aside from that you're looking at digging tunnels beneath to create this effect, not a quick job and a high chance of being discovered - the collapse may not be total, it could sink in parts.
This is perhaps more likely...but again still difficult, its all a matter of volume and position. If your island is in the middle of a lake with a dam then you're in luck, the lake is a small area so a rise of a few meters is much less water than if your island is in the middle of the ocean. Blowing a dam or diverting a river are possible solutions for flooding a lake. Otherwise you're looking at melting ice caps and flooding everything that lies as low as your island.
The consequences are important. You can't raise the level of the ocean to flood an island and expect the rest of the world to be fine. Equally if an island sinks the volume of water that is going to start rushing about has to go somewhere. Earthquakes at sea have been known to cause tsunamis.
All in all, if you're looking for a quick in and out job rather than a prolonged engineering project then you need explosives, a dam and your island to sit in the middle of a lake.