Network With Most Emmy Wins: NBC
Tracking the history of TV through the Emmys leads you through decades of only three networks, which is why NBC has by far the most Emmys of any TV distribution service (you can’t even just say "channel" or "network") with over 1,000, followed somewhat distantly by CBS with over 900. These are hard numbers for anyone else to catch up to — FOX has only been around since 1986, with its first nominations coming for The Tracey Ullman Show, followed by HBO in 1988 (which received nominations mostly for specials).
It's not surprising that, even though HBO had to make up for the other networks' 30-year head start, it continues to dominate the awards (it has over 120 nominations for the upcoming ceremony), with over 450 Emmys historically, including 99 wins just last year. It's by far the most successful cable channel, perhaps because subscribers pay directly for it in part for the prestige value associated with its shows being "not TV."
Most other cable networks are not so successful awards-wise — even AMC, which had runs with Breaking Bad and Mad Men, has struggled to place other shows. FX, the most successful basic cable network, was only nominated for 45 Emmys last year, less than half as many as HBO. (Historically, it's had hard time getting traction for any of its shows, with wins in the Best Miniseries categories but few wins for anyone not named Glenn Close.)
While streaming services appear poised to become the new HBO in this respect, attempting to deliver shows directly to the awards center of the brain. they, too, have yet to fully catch on, partially flopping with even its most prestigious shows: six for House of Cards and three for Orange is the New Black. But watch this space: In the next few years, this will be the "network" area set to start scooping up Emmys left, right and center.