In this day and age, whether or not something is available is no longer really a question. The only thing that is relevant becomes how much it costs to get it. In this case, the cost of shipping perishable goods to these "islands" on a daily/weekly basis.
Even if fish were sufficient, I don't think any of the inhabitants would appreciate eating fish and fish alone day in, day out. Non-perishable goods like crackers, chips, Pepsi, coffee, sugar, flour, etc. can be shipped in large quantities and cheaply (via barges). But stuff like eggs, milk, fresh meat, and other goods that can be held in storage for long periods of time without adversely affecting the quality and nutrition of their constituents need some other solution.
At any rate, they'll have an easier time at getting something to eat than the inhabitants of the International Space Station for sure :)
Just because it's in international waters doesn't mean it has to be in the middle of the ocean. If they're willing to accept being outside the territorial waters of any country but still in the Exclusive Economic Zone of one, then they can sit 24 miles off the coast from a port, and do weekly grocery runs with an ordinary vessel.
If they can't stomach being in somebody's exclusive economic zone (not that it makes much practical difference) then they'd have to be 200 miles offshore, making grocery runs a bit more of a pain but still possible. A reasonably fast ship will get you there and back in a day.
They should pay the government of Nauru to rent their sovereignty. They can keep the "island" above sea level by building a mooring tower and tie seasteads to it. Their territory can be the 200 mile economic zone of Nauru.
Also, there's lots of healthy stuff that can be kept in dry form. (Fruits, Legumes, and grains.) They will also have the ability to keep things frozen. Chest freezers can be highly efficient. Especially if they are not opened very often.
Even if fish were sufficient, I don't think any of the inhabitants would appreciate eating fish and fish alone day in, day out. Non-perishable goods like crackers, chips, Pepsi, coffee, sugar, flour, etc. can be shipped in large quantities and cheaply (via barges). But stuff like eggs, milk, fresh meat, and other goods that can be held in storage for long periods of time without adversely affecting the quality and nutrition of their constituents need some other solution.
At any rate, they'll have an easier time at getting something to eat than the inhabitants of the International Space Station for sure :)