Hfrhyu

I'm currently trying to set up an underwater race in my world—merpeople if you will—and I'm trying to figure out how their city would look. I'm basing their religion and lifestyle on native civilizations such as Inuits, or some Native American tribes, but I also want them to be sedentary. What materials available in the deep waters could be used to make houses?

Whalebone.

Those bones are big. They can be found on the ocean floor.

https://en.wikipedia.org/wiki/Whale_fall

Ancient Siberians made huts out of mammoth bones.
source

And people do make huts out of whalebone found on the beach.
https://www.pinterest.com/pin/223280094000002936

Probably there would be a lot more whale bones available to your merpeople than can be found on land. Plus bone huts offer fun writing possibilities. Besides whalebones, there can be other comparably large or larger bones used to make the buildings...

Assuming the mer-people can breathe through water, and mirroring Native American style homes, then it would be practical to form homes out of sedimentary rocks at the bottom of the sea. These would likely resemble the homes of cliff dwellers or adobe-style homes.

Most rock at the bottom of the sea is sedimentary; this makes it dense and hard. A creature advanced enough, with the intelligence of human civilization, could go through the stone age with tools available in the ocean, and using such tools chisel out homes out of the rock formations at the bottom of the sea. It would not be too far off from Native American style homes.

As far as the Inuits that you mention, the Igloo Construction would be a better shape for the water pressure of the ocean, but would probably be easier made as being chiselled-out into the stone (like the cliff dwellers), as mining rock only to re-form it would be difficult, especially because any type of mortar would be difficult (but not impossible) to make inside water.

Cegfault took the most obvious material: stone. Let me offer a couple more.

Coral

Houses could be grown through coral polyp husbandry. Officially, the material is calcium carbonate. Houses of nearly any size could be made through this method, though is would be slow. However, with a bit of engineering, walls, floors, and roofs can be built this way. Thick coral would be a reasonable protection from break-in.

Kelp and sea grass

Agriculture isn't just for food! Cultivated sea grasses could be planted, and woven as they grow to create sturdy walls. This solution would be excellent for tipi style domiciles. Or, if weaving walls isn't your cup of tea, you could cultivate personal groves of sea grass, then trim a path into the grove and rooms within the grove (think "corn maze").

Caves in ocean trenches

There are oceanic trenches all over the planet. The wonderful thing about trenches, is that they provides sides. Especially the deep ones. Caves can be hewn out of the sides. Hole households could be inside, so long as the local current isn't so strong that it can erode the cave mouth. This solution would work anywhere there was a reasonably steep, somewhat verticalish surface (the sides of volcanic mountains are especially prized for their central heating).

I like the Coral husbandry of JBH and carving out of cave/cliff walls. But I can add one more option.

A civilization may advance to a point where they need to use energy sources. Best energy source maybe geothermal vents and underwater volcanoes. this could provide energy for future technological development.

Now if your "merfolk" could harness this, they would build structures around these energy sources, maybe control the spewing lava to form the structure around it. So, you could create an igneous rock/obsidian structure.

You could augment these structures with other building materials. Unfortunately, where you would find these energy source may not be where you would also find coral or sea grasses. Sedimentary rock, such as sandstone would be readily available, or any other strong and tough rock that would hold up to currents, external forces and quakes that would most definitely accompany a volcano.

Maybe you can control the water chemistry enough to grow massive salt crystals to incorporate into your structures. This would be decorative in nature only and would be fairly difficult to do in this environment, but would make a great "temple" of some kind if your civilization advanced to a Aztec level society.