Tried to come up with counts for some games... and found it extremely difficult. What counts as an award? A nomination? nominations in multiple categories? Some random website?
Anyway, here were my leaders. My method was totally non-scientific, just counting up what was listed as linked awards on BGG, and trying not to double count a nomination and a win for the same award. feel free to add to this.
Dominion: at least 34 7 Wonders: at least 31 Pandemic: at least 28 Qwirkle: at least 18 King of Tokyo: at least 17 Ticket to Ride: at least 17