IANAL

But that is the issue, you as a developer and provider may not run afoul of the terms directly, but your users will.

Section 10 in the Basic Terms forbids accessing Instagrams Private API's.

And if that isn't being used and you are a user of Instagram are running afoul of Section 4 of those terms in soliciting, collecting, or using the login credentials of other users. As well as your naming being in conflict with Terms and trademarks.

Unless you're offering the service, through a reminder system with push notifications, you or yoru customer is probably breaking the Terms somewhere. You should probably have a lawyer verify that you're not running straight into a legal mess.