Contrary to what everyone is saying, Vayne does in fact have a terrible early game. Why is her early game so weak? Her range. Her range is terrible, and she doesn't do a lot of damage like Draven, or have poke like Cait. You can exploit this to no end to punish her. The best way to do this is to pick someone like Cait or Lucian into her for adc. They have enough poke to make the laning phase hell for her. Every time Vayne goes to last hit a minion auto attack her, poke her, whittle her down. When she's so low that fighting would be dangerous, zone her from all her cs by standing in or behind the minion wave. By the time the laning phase is over you should have a HUGE cs gap. If you manage to get a kill, even better. With so little gold she'll be no threat in teamfights and you can continue to bully her all game.
Because her early game is so weak Vayne needs a support that will help her survive: Nami, Sona, Soraka, etc. With someone to heal her she has more room to be aggressive and punish you. If her support cant sustain her, be more aggressive in lane.
Vayne's Q is her ONLY escape and way to avoid cc and skillshots. But because her range is so short I see a lot of Vayne player do this: They will Q through the wave and auto attack the enemy adc. This is VERY risky, because like I said, this is her only means of escape. If a Vayne EVER does this to you, beat her into the ground. CC her, auto her, chunk her down. Without Q she will have to walk back through the wave, with no escape, and you will be within AA range to continuously whittle her down.
In bot I usually play support. So, if you want to know what supports to play against her: Nami, Lulu, Leona, Naut. My FAVORITE support to play against her, and my main support, is Nami. If you play Nami into Vayne you have 100% chance of dumping on her. Why? Vayne CANNOT trade with Nami under any circumstances. EVER. Because Vayne's range is so short she will be within Nami's W range A LOT. Whenever Vayne is within range of Nami AA,W,AA her. This will take out a chunk of her hp, even if you cant AA her use W. Because Nami's W heals her, if Vayne AA's you, you wont lose any hp, every time Nami trades with Vayne she wins. So the only thing Vayne can do is get zoned. The other supports I mentioned have high kill pressure or poke, something Vayne never wants to deal with in lane.
Finally, against Vayne. Make SURE you get to level 2 first in lane. With something like a Leona support you can all-in her at level 2 while shes at level one and SEVERELY cripple her.
I hope you find this useful and good luck.