I don't think there's the one vegan stance on culling, but I can try to give you my opinion.

If we are purely talking about the ethics, the question always comes off as somewhat disingenuous to me. The vast majority of culled animals are livestock, and those animals were bred to be killed anyways. Whether a chicken is killed after six weeks to try and contain the outbreak of some disease, or killed after six months when it reaches regular slaughter age is irrelevant, as I consider both deaths to be avoidable and therefore unjust (especially considering that a lot of the diseases that would warrant culling an entire population are only an issue because of the terrible conditions those animals are being held in in the first place).

If we are talking about bees specifically, I'd consider culling a hive infested with e.g. foulbrood to be the correct thing to do - but I also consider it wrong to keep bees in the first place. Not culling the hive will inevitably cause the infection to spread to the native population, that likely already is weakened and has trouble to compete with the bee keepers hives.

[...] controlled culling of herds of deer is necessary [...]

There are lots of arguments about whether hunting is truly necessary and studies (e.g. [1]) showing that it might not be, but I'm not a scientist, don't understand those studies anyway and there am therefore not really qualified to argue either way. My personal issue with hunting (or culling in general) is, that I don't feel like it's being done to protect the healthy animals and the surrounding ecosystem, but for personal or monetary gain.

A farmer doesn't kill his H5N1 infested chickens because he is worried about the well-being of the native bird population, but because the chickens are now economically worthless and he is legally required to do so. The bee keeper similarly doesn't care about the native insect population, he will burn his hives because it is the only way to get rid of foulbrood. Both will simply turn around after culling their animals, start a new flock/hive and keep going. And hunters aren't biologists that are able to safely identify and exclusively shoot sick animals either. I suppose it depends on where you live, but if your average Joe is able to buy a hunting license and go kill animals with minimal training, you probably aren't exactly creating a healthy ecosystem. Instead, you got a monetary incentive for the state to sell hunting licenses and a bunch of people shooting animals for meat, trophies or just for fun, which is then again morally questionable and might, according to the aforementioned studies, counterintuitively even lead to an increase in overall animal population. Trying to get native predators back into the area is then blocked by those same people, because the farmer is worried about a wolf eating his livestock (loosing him money) and the hunter wanting to shoot a wolf. The media™ then runs a campaign about the scary wolves eating your dog and attacking your children, politicians fold over and wolves are being shot at, destroying any chance of the ecosystem recovering on its own.

most vegans would prefer the natural solutions

I'd say most vegans would prefer if animal farming just got banned. Given that 80% of all agricultural land is used to feed and raise animals, a lot of our ecological issues are directly linked to the animal agricultural industry. Giving this insane amount of land back to nature and just leaving it alone would probably do wonders to the general state and resiliency of the ecosystem.