Maps are essential in everyday life, but inherently inaccessible to visually impaired users. They must be transcribed to non-editable tactile graphics, or rendered on very expensive shape changing displays. To tackle these issues, we developed a tangible tabletop interface that enables visually impaired users to build tangible maps on their own, using a new type of physical icon called Tangible Reels. Tangible Reels are composed of a sucker pad that ensures stability, with a retractable reel that renders digital lines tangible. In order to construct a map, audio instructions guide the user to precisely place Tangible Reels onto the table and create links between them. During subsequent exploration, the device provides the names of the points and lines that the user touches. A pre-study confirmed that Tangible Reels are stable and easy to manipulate, and that visually impaired users can understand maps that are built with them. A follow-up experiment validated that the designed system, including non-visual interactions, enables visually impaired participants to quickly build and explore maps of various complexities.