SMGF does test Y and mtDNA and make the results available online in a publicly accessible database, showing pedigrees with the most recent generations (anyone born after 1906) privatized.
You can then search for your pedigree and deduce your results (with a little bit of effort), but SMGF does not send them to you directly, and your results will not be as fast as a commercial company. Here's an example with my husband's Y results -- he has no close matches, so I'm hoping more Turners will test
http://tinyurl.com/3aqnx7
SMGF tests a somewhat different set of markers than FTDNA, and you can't send SMGF results to FTDNA. However, many FTDNA project administrators maintain their own websites, and some of those will accept results from SMGF as well as other commercial companies. There is enough overlap for useful comparisons.
SMGF is beginning to collect more samples in various places around the world. There's a map of their current progress here:
http://www.smgf.org/maps/collections.jspx
African Americans can make a big contribution to the research project, even if they don't know their exact origins in Africa. For this type of research, it's helpful to have participation from extended family members (siblings, parents/children, grandparents, aunts/uncles, cousins). For example, if a cousin on your father's side matches some of your autosomal markers, you can tell which side of the family they came from. If you test all by yourself, you can't tell. That's what makes autosomal research so tricky, but it's the next frontier.
co-author (with Megan Smolenyak) of "Trace Your Roots with DNA"