Per my cursory research, the record in a season is 7, done 3 times.
1990: Mark Langston & Mike Witt, Randy Johnson, Nolan Ryan, Fernando Valenzuela, Dave Stewart, Terry Mulhollan, Dave Stieb
1991: Nolan Ryan, Tommy Greene, combined 4 Oriole pitchers, Dennis Martinez, Wilson Alvarez, Brett Saberhagen, 3 combined Braves pitchers.
2015: Chris Heston, Max Scherzer (twice), Cole Hamels, Hisashi Iwakuma, Mike Fiers, Jake Arrieta.
1991 would have had 8 as Mark Garnder of the Expos no-hit the Dodgers for 9 innings 2 days before Dennis Martinez perfect game (mentioned above), but the Expos failed to score when he was out there.
None of the years above had 3 no hitters by May 6. Only one of the no-hitters above (Langston/Witt) was by that date.
Offense is down. Could we see eight this year?