what MLB team holds the record for most hits in a game without scoring a run?

Answer 1

14 hits is the record for no runs scored, held by the New York Giants in the NL (Sept 14, 1913, against the Chicago Cubs).

14 hits is also the record in the AL, held by the Cleveland Indians (July 10, 1928, against the Washington Senators)

Answer 2

In a nine-inning game, the record is 14 by the 1913 NY Giants:

http://www.retrosheet.org/boxesetc/1913/09141913.htm

The extra-inning record is 15.

Answer 3

14 hits by NY giants of 1913