Clothing store access in Pokémon X & Y

Here is a picture from (what I think is) the official Pokemon X&Y Guide.

Every little thing you do in Lumiose City adds to your style. Stylishness brings all kinds of rewards, such as meeting various characters, getting discounts, and unlocking facilities and fasions.

Good luck fellow trainers :)

The NPC outside the boutique says:

Being stylish is about knowing everything there is to know about Lumiose!

This led me to believe that if you visit all of the main attractions, even if you don't do anything in them, you unlock the boutique. I used the connecting pathways to get through them all systematically, but I suppose the taxi service would work too.

The easiest (although arguably lame and kind of "cheaty") and probably cheapest way is to buy a bunch of Premier Balls.
Make sure you buy them one by one. It is not necessary to exit the store or the buy dialog, you can simply select the Premier Ball and mash A.

I personally needed roughly 120 Premier Balls to unlock max style (which can be checked, among others, by trying to buy a Mega Stone in the Stone emporium; if you have max style, the price will be 10.000), but I did some style-raising actions before that, so I assume 150 should do if you start at 0 style.
Don't believe the reports on the Internet which say you need 100. They're based on wrong observations because the people trying it had done more things to raise their style before starting to buy Premier Balls.

Note that Premier Balls only become available after you beat a few gyms (I believe it's 4, but I'm not sure). This will also work with any other item, but Premier Balls are the cheapest and at the same time most useful ones. I believe you have to buy from a Lumiose exclusive shop though, the PokeCenter mart won't do.