How Does the Water Tribe Dye Their Clothing?
There are a few "realistic" ways for the Water Tribe to have acquired the blue coloring for their clothing:
Trade with Kyoshi Island
I headcanon that the Water Tribes originally imported indigo dye from Kyoshi Island, prior to the war (for the North) and the raids (for the South). Kyoshi Island is partially inspired by Japan, which is famous for its high-quality blue indigo.
Crowberry and Arctic Blueberry Dye
Both of these berries are native to the arctic. On their own, both berries produce purple dyes. However, if you mix these dyes with iron/rust-infused water or vinegar, they can become beautiful shades of blue.
Woad Dye
Woad is a plant that grows in subarctic climates like Southeastern Sibera. In the ATLAverse, they would probably grow using greenhouses in the NWT. Traditionally, the blue dye was created by fermenting woad leaves in a mixture of water and stale urine.
Or, alternatively, you could just ascribe their abundance of blue dye to a fictional animal or plant. My headcanon is that they dye their clothing blue using a poisonous-when-eaten plant to ward off large animals. It also has the added benefit allowing them to blend in with the sky when they stand along the horizon; this would be advantageous when hunting.