Yes, you do need access to the internet to use FanDraft at your league's draft.
FanDraft is a web application, so having an internet connection is required.
For leagues that do in-person drafts, where only the Admin/Commissioner will be running the entire draft (utilizing "Commissioner Mode", that person will still require an internet connection at that time -- as again, FanDraft is accessible from the internet.
But I'm Holding an In-Person Draft somewhere remote without Internet
If you're holding your draft party somewhere remote, like a cabin in the woods or a boat in the middle of the lake, you may not have wired internet. However, it's VERY likely you have a cell phone connectivity. Nearly all phones and plans these days will have the capability to set up a personal hotspot. For instance, if someone in your league has an iPhone, you can easily connect your laptop directly in and have internet for your draft.
Below are resources with instructions on how to do this: