Can I get banned for using LSI (League Summoner Information)?

Maybe eventually, but probably not.

Strictly speaking, it is a third party program, so Riot can't approve it. Third party programs are updated all the time and Riot doesn't have any control over what changes are made, so they can't explicitly give the OK to any of them. From a recent post on the forums:

We will not be maintaining a list of applications that might be approved or denied, because honestly, we’re just not ever going to be able to continually audit every service or function for player security, especially in this ecosystem where devs so frequently improve and expand their apps. And just to clarify, Curse is not "100% approved" specifically either, though it is okay to use. We only spoke about it specifically by name to clear up confusion around the fact we did initially prohibit its use, but we want to make sure it's clear that it is not "100% approved" as we do not make that statement about any particular app.

However, in this specific case it's worth noting that the people working on LSI stay connected very closely with Riot to make sure that their program doesn't violate the Terms of Service and won't fire false flags for the methods Riot uses to detect scripts and the like. In addition, the player ranking information is freely available on several websites for the same reason LSI can find it: Riot has a public-facing API, which wouldn't offer this info if they didn't want it to, especially given how long it's been up and how established some of these sites/programs are.

So if in the future changes are made to LSI without consulting Riot, it's possible that using the program will be a bannable offense. But at the moment it seems that's not the case.