On the second floor in any Pokemon center there is a door with in attendant to the far right and she will ask what you want to do. (all of these options involve wi-fi) Then do GTS (Global Trade Station) and you can search for Eevee that is up for trade or deposit a Pokemon to trade for an Eevee. You can specify what level and gender if you want to, but to search for one in the GTS you have to have seen one in-game.
If you search for an Eevee that is already up for trade, and you don't have the Pokemon that person is asking for in exchange, but you know you can obtain it, go get it as soon as possible. You will probably have more time than usual to do it though, since everyone is playing X and Y.
IF YOU DO deposit a Pokemon in trade for Eevee I highly recommend you put a newly hatched clone of your starter. If you do not have a Ditto they are found in Giant Chasm. When you deposit a Pokemon, you have to leave it there for a while until someone finally trades with you. As soon as you enter GTS when someone has traded, a pokeball will fall from the sky and out comes your Eevee!
The GTS is fairly simple and easy to follow, I mean the main menu of it has three buttons so...
Hope I helped, my source for all this is experience.