You can only get fruit, vegetables, and seafood by trading with the islands once they unlock at level 29 and you restore the Port.
Frutus Isle (unlocks at level 29):
- trades peaches, watermelons, and plums
Olivia Isle (available for exploration at level 35):
- trades grapes, olives, and key limes
Tropica Isle (available for exploration at level 41):
- trades bananas, coconuts, and pineapples
Fishermen's Isle (available for exploration at level 52):
- trades fish, shrimps, and lobsters
Your chances of getting different goods from the islands vary. However, you can load ingots onboard while preparing ships to boost your chances of receiving rare products and treasure chests. Ingots are twice as efficient when you use the Favorable Voyage booster at the Laboratory.