Sampling a uniformly random satisfying assignment
Yes.

(backup links in case that one goes down: 1 2 3 4)

Back up reference, in case all those links go down: Bellare, Mihir, Oded Goldreich, and Erez Petrank. "Uniform generation of NP-witnesses using an NP-oracle." Information and Computation 163.2 (2000): 510-526.