LoL uses Riot Points as a currency for their store where you can various stuff. The rate is 650 RP for $5 which is enough to buy some champions or skins. If you're interested on the newer Champions or more popular skins however the price is a bit higher.
The first things you can see on the store tabs are Champions and Skins, this is where the meat of the store would be. Because players can only access a limited number of random Champions each week, they will eventually have to unlock these Champions if they want to play them at any time. Players can buy Champions with IP(in-game currency) which takes quite some time to farm but perfectly obtainable. Skins however can't be bought with IP and players need to spend real money for them.
Skins are of course an okay thing to be sold on the store since it is purely cosmetics and doesn't offer any stat change. Selling Champions are also okay because it is quite properly balanced and players can just unlock any of them with in-game currency. New Champions costs more IP so it would force some people to buy it with RP, but as this is purely optional I really don't mind this kind of practice.
For cosmetics stuff, aside from Champion skins there's also Ward Skins and Summoner Icons. Just like Champion Skins, these are purely cosmetics and will not affect gameplay. In my opinion the pricing is a bit high for just cosmetics stuff but it is still acceptable.
A few people have voiced their concern regarding the next two tabs: Runes and Boosts. Runes gives various passive bonuses to the player's Champion while Boosts well...boosts the player's IP income or experience gain. The concern are that some people might be able to Pay to Win in the low level by using IP boosts to gain plenty of IPs and use these IP to get a full set of Tier 1 Runes and make themselves stronger because most people will never buy Tier 1 runes.
While at first this sounds plausible, this problem is actually really minor and irrelevant in the big picture. No one will want to create a new account and put real money for boosts multiple times. And even if they do, this will only be effective for a very short time until that player reaches a high enough level where Tier 1 runes doesn't matter anymore. And again even if you want to consider the small amount of time that this can happen the Tier 1 runes aren't potent enough to actually make a difference since low level players have a very limited rune slot anyways. So in short, no...there's nothing P2W about the Runes and Boosts(and no, leveling up faster doesn't make it P2W it will actually give you less number of matches to familiarize yourself with the game and will quickly match you up with higher level players that have more experience than you).
Now we move on to the "Others" tab which contains region changes, name change and additional rune pages. These are all optional stuff that no one in their right mind will think "Pay to Win" so I will not talk more about it.
In conclusion, LoL's business model is of course as you might have guessed: not Pay to Win. The store contains mostly cosmetics stuff and stuff that actually affects gameplay are available for in-game currency making most purchase a case of time vs money, which I really don't mind. Can a person play LoL without spending any money on it? Yes, of course they can and they will not feel that they are weaker or less competitive because of it.