Is it possible to have a satellite (natural or artificial) in orbit around a planer which is tidally locked to its sun?
Yes, kind of.
"Geostationary orbit" in such a case would be an orbit around the Sun at the Lagrange points 1 or 2.
(distances in the image are not in the right scale, the points L1 and L2 are much closer to the planet)
Such a satellite would always be above the same spot of the planet. Its orbit around the Sun would last exactly as long as the orbit of the planet. Its orbit around the planet would be "infinitely long", so the satellite would not move in the planet's sky. Therefore it would work just as a satellite in geostationary orbit. As for Earth the points L1 and L2 are about 1.5 million km away.
EDIT: As others have pointed out, orbits at L1 and L2 are not completely stable, so orbital adjustments would have to be made. The James Webb telescope is at the L2 point A space elevator between the hypothetical planet and the L1/2 points would be possible, and be just as good as a space elevator in geostationary orbit (if you can make a long enough tether).