Classic Bookshelf: "Electronic book" version of 18 tales, with online reader.
ClassicReader.com: Text of some of the fairy tales.
eBooks Cube: Andersen's Fairy Tales: Text of 18 stories.
Hans Christian Anderson Fairy Tales: Collection of full text stories for children.
Learn Library: Text of 18 fairy tales.
Project Gutenberg: Andersen's Fairy Tales: Plain text version.
Reading Room: Full text of some tales available via Java applet.