As I understand it, most large conventions attach a membership number to each attendee. If GenCon does, that number can be used in on-site voting, to ensure each person only votes once.
This does not solve the problem of finishing the awards before the ceremony. If one wants on-site voting, one might have to accept that the physical awards get sent out after the convention.
There is another reason to question spreading the awards out too much, though. It is a question of mandate.
Perhaps one of the bigwigs could tell me - how many people participated in the Ennies voting last year. If I recall correctly, it was an embarassingly small number. Down in the three digits for a site that claims to have 20,000+ registered members.
So, now, the largest gaming convention around has awards given out by a few hundred folks, most of whom probably don't even attend the convention? Well, if you keep the scope of the awards down to something that the voting group really claims to know and care about - d20 games, then it's okay. Spreading out the mandate to cover other rpgs is weak, but perhaps supportable. Spreading out to cover other sorts of games where are rarely even discussed on these boards is downright presumptuous and shabby.
EN World has a solid claim as top D&D fans. A good claim as top d20 fans. We can even reasonably say that we're collectively RPG fans in general, though discussion of other systems here is relatively minor by comparison to other sites. We've got no real claim to collectively being afficionadoes of anything else, though.
Honestly, if GenCon wants braoder awards covering every sort of game seen there, it needs to institute it's own awards. It would need something like the World Science Fiction Convention's Hugo Awards, voted upon by the people who attend the convention.