I know about a few of the archives, Gutenberg,University of Arizona, the link thru Georgia Seitz.
I did not know about the Library of Congress. There are a small group of tatting books there and some I have not seen before. Check it out: http://www.archive.org/search.php?query=tatting
Once you click on your choice you can choose to read the book online,good for deciding whether it's worthwhile saving or if you already have it. Or you can save it to pdf.