They are usually depicted as not very sentient beings, I cannot imagine them say driving and maintaining farm machines, growing crops and animals, then you need the industry to provide fuel and parts for these farm machines.
Well, the original zombies of legend were slaves working on plantations in Haiti.
The legend also says that they were controlled by necromancernecromancers (voodoo practitioners), so the zombies would only do what they were told to.
Zombies are like flesh automatons. Give them simple commands and they will execute those. If you have enough zombies, even if each one is doing one simple thing, the phenomenon of emergence happens. The zombies are then able to build and operate very complex systems, just like ants in an anthill or bees in a hive.
Granted, you need maintenance here and there. But a single necromancer should be able to handle it. They don't even have to be present in the zombie lands throughout the year. If they pay a visit every once and then, they can adjust what needs adjusting and ensure that the zombie society thrives.
In such a system, a few zombies could efficiently take care of farming to feed all the other zombies. They could raise cattle to satisfy their daily intake of brains. The rest of the zombies would work to bring money into the zombie continent so that they can trade for whatever it is that they and the necromancer need.
Zombies are perfect for jobs such as management, accounting, IT and customer service, since these are areas where having a negative charisma score is part of the job description and a living person would have to leave their soul at the office's door anyway. In this day of remote working, zombies would be even more competitive against the living in the job market.