Can vegans eat honey? Is it against their moral code or something?

Nope, honey is made by bees and is therefore an animal product

Yes, they can't eat honey.

As you said, it is against their moral code, as at their point of view, beekeeping is another way of animal explotation, and they don't eat anything that means animal explotation, such as milk, cheese, honey, eggs, etc.

Vegetarians don't eat meat, but vegans don't eat meat or animal products. I think that they wouldn't because honey is still animal produce