Islands get running water through water wells drilled into the ground or through cisterns.
Some Islands use the saltwater from the Oceans after it goes through desalination.
If someone is rich enough then they can have a personal desalination plant onsite to desalinate the saltwater into fresh water.
Luxury Island resorts use very efficient desalination to access fresh running water.
Some islands make use of groundwater for public water supply, meanwhile the groundwater heavily depend on weather conditions to sustain itself.
Recharge usually occurs in the peak wet season months, and it is key that the monthly rainfall is above the threshold of 195 mm monthly.