***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.
Get the Eevee from Bill and go to the trading place in the Pokemon Center.
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.
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!