Based on the description of the conditions that these people live in I would say that there are several ways of sustaining a population on a floating island.
Agriculture on a massive scale is not possible due to lack of space but what if your floating landmasses have cave systems which could be used to grow mushrooms? Citizens of these floating cities would need to dig the cave, maintain it and grow mushrooms in a system similar to hydroponics. Mushrooms can grow easily but my family used to grow them on sacks of wood and sawdust. People living in your floating island could just stack these things for a rather large farm.
Are hanging gardens a possibility? They could perhaps grow fruits on a massive floating garden which hangs from the sides or bellow the island. A complicated system for harvesting will most likely be required but plants could grow "upside down". They would have access to plenty of sun and water from the air. Nitrogen supplies could be a bit problematic for the plants, but your citizens would need to get them somehow and apply them to the plant roots.
Hunting is an option, depending on what kind of wildlife inhabits their world. Are flying whales or any other large flying creature a thing in your world? If yes those floating ships could be used for hunting expeditions. They could hunt large flocks of birds with massive nets.
Raids on the ground or the ocean. I know you said going down to the ground is forbidden, but a man's gotta eat. Perhaps a blessing to a special cast of hunters could be given to venture on the ground to forage for supplies? Sending flying ships to hunt for fish would not violate those rules. Expeditions to tree-top farms?
Insect farms. Maggots will grow on rotting corpses and you mentioned zombies being a problem in the world. Kidnapping them, putting them down and using them for maggot fuel could perhaps provide some meager sustenance to your citizens.
Your floating population can easily get water from air by gathering morning dew and rainwater. They would, however, not have enough water for luxury and would most likely have terrible hygiene.