No. ENIndexer is a downloadable application that runs on Windows and Mac platforms. No Internet access is required to use it.
However, the User Guide and Quick Start Guide are maintained online so you will either need an internet connection to be able to see those files, or you can download them once to your machine and access them at any time locally.
You will need an internet connection when you register your software, but you can register from any computer if you don't have a connection from the machine you are running ENIndexer on.