***Actually, it is possible to obtain if traded from Ruby, Sapphire or Emerald or that Larvitar.*** Apart from the Larvitar as mentioned below or Mighteyana, Crawdunt and Sableye (which can only be obtained from RSE), an Umbreon can be obtained as well although it is a bit of a hassle. It requires two Gameboys and a copy of Emerald.
Step #1
Get the Eevee from Bill and go to the trading place in the Pokemon Center.
Step #2
Trade to Emerald because the clock feature exists there. Check the time max out happiness and level it up (make sure its night time). You now have an Umbreon.
Step #3
Trade it back to Leafgreen.
As to your question about whether Dark types ***EXIST*** in Leafgreen, no. It's only possible to obtain a Dark type through trading or as said in other answers, beat Blue then go to the Sevii islands to catch a Larvitar.
Hope I Helped!
-It's a secret!