Samoa international Vui, 24, joined Warriors in December and has played eight times this season.
Full-back Protheroe, 20, and back-rower Thomas, 23, have made four and 15 appearances this term respectively.
Centre Bedlow, 21, came through the Sharks academy and has played in eight games.
Bristol were relegated back to the Championship earlier this month after just one season in the Premiership.
Bristol have signed Worcester Warriors flanker Chris Vui, Gloucester pair Mat Protheroe and Dan Thomas, and Sam Bedlow from Sale Sharks.