It all depends...

If there's a LOT of vegetation, trees (other than mangroves) and such, then there has to be quite a bit of fresh water, somewhere. You might have to dig for it, use seeps until you find better sources. Finding the lowest path the water would take back to the sea can't be far off.

I don't carry enough plastic for a solar still in either EDC or PSK, so that's outside the rules. You'd be lucky for one to survive a storm or two, anyway.

People have survived for months drinking only the blood of sea turtles and the occasional bird. Blood is far less saline than seawater, and will keep you alive, if you can get enough of it.

But.. there's a reason they call them "desert" islands, and if any large island is unpopulated, there's probably a reason for that as well. There may be enough to sustain one person but not a community... but there may not. You can't survive every possible scenario, and sometimes "no fresh water" is really NO fresh water.

Some new reverse-osmosis desalination systems use amazingly little energy, and may actually change the definitions of what's "inhabitable" or not... but it's nothing you're going to have in a pocket.