As someone who enjoys lane bully adcs such as Caitlyn and Draven but also a fair bit of Ezreal and Vayne, I feel I've got a good bit of experience on the matter of Vayne being OP or not.
In my opinion, she's not.
Vayne's strength lies in her slippery dueling potential, in the early game she needs to farm, level 2 all-ins from Leona, pressure from Blitz or Thresh and constant harass from Draven or Cait will hinder that so much it's not even funny. Notice that I don't add Lucian to the list, because even though he is a lane bully, his abilities push the lane easily, causing a very advantageous situation for Vayne. Also Vayne's catchphrase "Black Magicks" and that hurts the poor man's feelings.
If by level 4 you're a good bit ahead in farm you can generally consider the lane won, keep the pressure up and be careful once she gets Cutlass, as this is where extended fights can quickly be turned around.
In the midgame, Vayne shines as the typical ADC. Her DPS is great, she's good great chasing and teamfighting and it's just bad news if you can't catch her in time. This is by far her most volatile stage, this stage makes or breaks Vayne's endgame potential. Focus her, kill her. Plonk down pinks and oracles and for the love of god don't splitpush alone unless you've got the whole map lit up like a christmas tree or have a VERY good escape plan.
In my experience there's a few major iterations of late-game Vayne, and it seems to mostly depend on playstyle and how well the Vayne did early and mid game:
The Assassin: This is the most high risk, high reward playstyle and usually played when Vayne got fed early/mid and did very well in general. When 5 to 6-slotted Vayne is able to single a target out and smash them before they even realise what the fuck just happened. Usually played by the most mechanically skilled and confident of Vaynes. CCing may help against this Vayne, but the chance of Cleanse or QSS being in her arsenal may make that redundant. Never travel alone.
The Teamfighter: Same as any adc, but with a scary amount of chase and damage. This Vayne did average to decently well throughout the game and is not confident enough to roam for a pick, therefor she sticks with the team and tries to launch skirmishes which are quickly won with Silver Bolts and Tumble. Should a full-blown teamfight break out, these Vaynes pop ult and dart around the teamfight hitting whatever they're comfortable with hitting until whatever they're hitting dies or they die themselves. Handle as any other adc: with CC and focus.
The Splitpusher: Vayne's dueling potential is quite crazy, but this is arguably the weakest playstyle which Vaynes often wrongly force themselves into after a bad early and mid game. They'll splitpush as much as possible, dueling whoever comes to stop them. Keep in mind that these Vaynes are usually underfed and underleveled and are trying desperately to catch up by shoving the nearest lane. Avoid 1v1s, engage with a teammate and keep her from the farm. Turn it into a mini laning phase. Or simply engage their team 4v5 and get a free fight.
That's all I've got on the Night Hunter, keep in mind that this is my personal experience and opinion. And so it will probably differ from most.