You can harvest blood from corpses that are already dead, and there are plenty lying around. Some are more common than others but it's possible to complete the quest without killing a single mer yourself.
If you're willing to cheat, the UESP Wiki has a list of places where pre-dead corpses can be found. If you want to look for yourself, the best places to look are dungeons with their own built-in protection of their own. Bandits camping out in ancient ruins often get killed by the resident draugr, and falmer are often killed off by the dwemer machines.
You can also take advantage of things like vampire or dragon attacks in towns that have a sufficiently large mer population.
Of course, if you just take a bunch of radiant quests to clear out bandit lairs, you'll inevitably end up killing plenty of the correct races just by playing. IIRC the only one I had to actually look around for was Altmer, as mer bandits tend to be mostly Bosmer and Dunmer.