The Vegan Society say they should not. I have written to them to complain about what they have on their website. They talk of beekeepers killing bees!
Some Vegans will eat honey, so long as no bees were killed in making it. Most won't because they won't eat any products that were produced as a result of the expolitation of animals.
The simplest reason why honey isn't vegan is by definition. The term vegan was coined by Donald Watson in 1944 and was defined as follows:
Veganism is a way of living which excludes all forms of exploitation of, and cruelty to, the animal kingdom, and includes a reverence for life. It applies to the practice of living on the products of the plant kingdom to the exclusion of flesh, fish, fowl, eggs, honey, animal milk and its derivatives, and encourages the use of alternatives for all commodities derived wholly or in part from animals (Stepaniak).
Vegans will not exploit an animal for its produce, therefore they will not eat eggs or milk, despite the fact that the animal has not been killed in order to produce the eggs or milk. Therefore I would say that vegans shouldn't be eating honey.
Furthermore, the vegan position on honey is definitive. Honey was prohibited for use by vegans according to the 1944 manifesto of the British Vegan Society (veganism's founding organization), a position consistent with the requirement for full (vegan) membership in the American Vegan Society since its inception in 1960.
People are free to make their choices, and I would defend their right to choose, but I do wonder at the selective logic applied by some folk.
When vegans eat their fruit & veg, are they not exploiting the insects that pollinated those plants? And the nutrients that made the plant growth possible came from the soil which was made productive by decaying animal matter and the work of micro-organisms, so when they tuck into the potato or cabbage aren't they exploiting the labours and deaths of animals?
I eat, therefore I exploit. Can't escape it.
So if the conscience is governed by just the degree of exploitation, I'd say the Vegan Society should review its stand on honey.