was it the dunmers? or other race?

Bugs.

Impossible to say.
But like our own ancestors used what was available to them I would think that most cultures in Tamriel that had access to or fought giant bugs from an early stage on would have used their chitin.
Dunmer obviously, but also subterranean races and possibly Black Marsh, who knows what lurks there.

Interestingly, not the Dwemer, though they obviously had access to giant bugs both from underground and Morrowind.
But it fits the Dwemer insular and almost cloistered way of thinking.
Not much they did seemed to have to do with the outside world as it did navel-gazing and being.. Dwemer.

Chitin is like the shell of a bug.

Who made the first Chitin armor? Dunmer would be my guess

It's possible that they did use chitin, just that it didn't get preserved as well as their metal artifacts. This is somewhat discredited by the overabundance of rugs and tapestries in Skyrim.

Woven from metal thread.

Elves, as a rule, seem to like chitin as a light armor. Both the Dunmer and the blind Falmer use it. And while it doesn't appear that the Altmer use it, I think its all but conclusive that the Bosmer probably use chitin in armor as well.

Interesting to think about what did not survive all those centuries.
We have stone chairs and beds, but not bedding or covers and I dont think Dwemer were so above the material they did not use ergonomics to ease strain on their bodies.

I like seeing all those 'ruined books' in Skyrim Dwemer ruins, it makes me think that the few intact books found in them were left there by later explorers and maybe someone like Calcelmo would love these ruined and authentic Dwemer books for even the scraps of writing they contain.

We can learn a lot about the Dwemer by what survived of them, but I think equally so if we think about what did not survive but what fleshy mortals universally use.

We dont find Dwemer clothing so that was probably organic in nature and has long rotted away.
Perhaps Chaurus silk?

If you've got giant bugs, it makes sense to put it to use- Chitin is a strong, lightweight material, especially in the form it takes inside arthropod shells.

Yes, and perhaps the Orsimer also have some sort of chitinous light armour made from their centipede livestock.
I would love to see that.

This one gets it.