For anyone reading this who isn't aware- horses can't be sick without medical intervention.
If they eat something that disagrees with them for whatever reason, they can develop a condition called colic, which is very painful for the horse and often requires expensive veterinary intervention to resolve and in some cases can lead to the death of the horse.
Horses are adapted to a really low sugar diet- sparse steppes grazing- which is why they can thrive in places like moorland with very little human intervention. Even the sugars in the very green grass which is more developed for feeding cattle can be too much for them- this can cause them to get painful metabolic conditions such as laminitis, which the owner can usually manage- but not if lots of passers by are feeding high sugar foods like apples (for a horse, any fruit and veg is high in sugar as it's not a part of their natural diet).
Hand feeding can also cause horses to become aggressive, e.g. if a footpath runs through their field.
If you want to interact with horses, their are many riding schools and petting farms which will allow you to do this (although many won't let you indiscriminately feed the horses for the reasons described above).
Horses are expensive animals to keep and they sort of cross the line between working animals and pets. In some cases, if they become unable to work (e.g. due to laminitis) they may end up being PTS if the owner is unable to manage this in their current set up.
OP, could you run electric tape across the gate, or wherever they are climbing in? It will be a PTA for you, but might discourage people climbing over, and if the habit gets broken for a few weeks, you may be able to get rid of it in the future.