The latest (2000) edition of the American Heritage Dictionary includes "website" as a single word (though it shows the two-word version as still preferable). This follows a frequent pattern of linguistic evolution, as new compound words often start as multiple words, or hyphenated, but eventually become a single word. Looking at books and newspapers from around 1900, you find things like "base ball" and "New-York", which give them an interestingly archaic flair. This is exploited by The Onion in its satirical Our Dumb Century book full of fake newspapers from the 20th century, in which early entries have bogusly hyphenated words like "news-paper".
Another point of concern is whether to capitalize "Web site". The "Web" part would seem to take a capital letter if referring to the Web as a proper name for a specific named entity (short for World Wide Web). Similarly, "Internet" is capitalized to indicate the specific network of that name, while a lowercased "internet" refers merely to any interconnected set of networks, which might not be the Internet. By this standard, a "web" (lowercased) might refer to a set of hypertext documents (in an intranet, for instance) that may or may not be part of the Web (the global one). However, the Web and the Internet are now largely viewed as generic parts of the world's infrastructure, like the telephone network (which isn't uppercased), so the increasing use of lowercase letters for these names is understandable, even if bothersome to old-line purists.