The private alpha release is now available: http://alpha1.simplytestable.com/.
You can run full-site HTML validation tests on any site that has a sitemap.xml file; it's from the sitemap that the URLs to be tested are retrieved.
There are a few known issues which translate into matters I'll be addressing this coming week:
the progress and results for large sites (2000+ URLs) will tend to not load more often than not
individual tests (that are part of a full-site test) that fail will tend to prevent the rest of the test from progressing
There are some improvements to be worked on also, including:
adding support for compressed sitemaps
retrieving URLs from a site's RSS feed if no sitemap is present