You don’t need to integrate with the LOL API to get item tooltips. Several sites offer javascript which will turn item links into the type of tooltip you’re referring to. Put the javascript into the head of your site and item links will be converted automatically.
Be careful which one you go for though, third party javascript running in your page can do pretty much anything.
ZAM offer javascript for a number of their sites including WoWHead and DestinyDB but haven’t ported it across to LOLKing yet.
If there’s a particular site you trust that has tooltips enabled check their page source to see whose code they’re using.
LOLBuilder looks to offer one option - http://tooltip.lolbuilder.net/
As does Arcana Gaming http://lol.arcana-gaming.com/en/Tooltip