Keep in mind that 1"x1" S4S stock is really just 3/4"x3/4" square doweling, so you are often most likely to find it with other doweling stock. However, such stock is often no more than 3-4ft in length but can be had in a number of species (especially if ordering online).
If you need any significant quantity of such stock, it is usually far cheaper to create and mill it yourself from larger stock. You can do so entirely with either a tablesaw or bandsaw, but if you want really smooth and consistently thicknessed stock then cutting the stock slightly oversized and feeding it (twice) through a thickness planer will provide you with all the stock you might possibly need and all of it will be of a single uniform thickness (provided there are no large changes in humidity or board moisture levels during the milling process).