Several things come to mind:
1. Limited water doesn't mean NO water.
2. Surface water may be very limited or non-potable. Qatar is something like 55,000 square KM with NO surface water (only country on Earth with NO surface water whatsoever in fact)...
3. Subsurface water / aquifers could well provide adequate water if they can be found and drilled.
As various posters have pointed out so well, the possible reasons for limited water are many and varied.
Point is - it's definitely NOT out of the realm of possibility to have a 40-mile-long island with limited water, even in a tropical clime.