It is explicitly illegal for non-humans to have wands (house elves especially, but also goblins and such). I think it's in Book 4 that Winky the house elf is suspected of using a wand, and is severely punished. They are powerful enough to use magic without; if they had wands, it would be much harder, I think to continue to subjugate them as a slave underclass. Wandless magic is supposed to only be accessible to very powerful wizards, so the fact that elves and goblins can do a bit (with no training!) is a testament to the fact that they would be very powerful indeed if they could more easily channel their magic.

It's also wild that a number of magical nonhuman species are apparently interfertile with humans, including giants (Hagrid is half-giant), goblins (Flitwick is, I think, 1/4 goblin), elves (there's a half-elf in Fantastic Beasts), and veela (Fleur is part veela). Hybrid wizards are, apparently, allowed wands under certain circumstances, though perhaps only if raised "as human." The whole universe is... somewhat broken, in this regard. Some really difficult ethical quandries come up when you notice that nonhuman wizards are essentially subhuman enough to enslave or otherwise treat like animals (see: giants, centaurs, merfolk), but are, at least sometimes, sexually exploited as well, resulting in hybrids.