My very elderly Mother left her home, in a small town (pop 800's) to live with one of my siblings "temporarily", for the winter. A full-timer vanning friend of mine was looking for a place to park for a while and after they met each other, Mom was thrilled to have my friend park in her driveway as a security measure. She agreed to have her hook up an extension cord to an outside outlet and my friend pays the electric bill.
The van rarely moves, as it is a small town and everything is within walking or bicycling distance (including a small, 24 hour gym with shower!). Nobody has bothered her, so far, since September. I made sure, though, that she met the immediate neighbors and they knew the situation. She also lives very quietly -- no generator, no loud music, etc., and keeps a low profile.
I think it depends on the town. Fredericksburg is a big tourist area and may not welcome boondockers. On the otherhand, they may welcome your dollars. Give it a try! Worst that can happen is you are asked to leave.
Best wishes.