Yep, I did. Actually, as an update, I found a few places near me.
Vanderbilt Autonomic Dysfunction Center in Nashville, TN (but has a tedious referral process) and Sentara Neuromuscular and Autonomic Center in Norfolk, VA both perform the test. Other places a bit further away that also perform QSART are Cleveland Clinic, and University of Pittsburgh Medical Center.

Their respective websites are-
Vanderbilt:Vanderbilt Autonomic Dysfunction Center - QSART Testing - Vanderbilt Health Nashville, TN
Sentara: Sentara Neuromuscular and Autonomic Center | Sentara Healthcare
Cleveland Clinic: Quantitative Sudomotor Axon Reflex Test (QSART) Additional Details | Cleveland Clinic
University of Pittsburgh: Autonomic Testing | Department of Neurology | University of Pittsburgh

I've also heard that Mass General Hospital in Boston, MA, and the Mayo Clinic (based in Rochester, MN, but with satellite locations in multiple other states) also perform the QSART sweat test.

I'll update this thread as I find more information regarding specific doctors that perform the test and how they've been reviewed, etc.