Nowadays, all supermarkets sell non-dairy milk such as oat milk, soy milk, or almond milk. Therefore, it makes it very easy to go dairy free. The same goes for yoghurt and cheese, however vegan cheese can be harder to find in Sweden. Some supermarkets sell it, but if you can't find it, Goodstore (mentioned in the Store section of my Find Restaurants page) on Södermalm always sells a large assortment of it. Vegan meat alternatives can always be found in the freezer sections of supermarkets such as Coop or Ica. Many brands such as Ben & Jerry's have recently developed vegan options as well, and McDonalds now offers a McVegan on the menu, which tastes amazing.
Being vegan is not difficult at all in a big city such as Stockholm, with a large variety of alternatives.