It makes full sense that the FBI would get a report and share with local law enforcement (members of JTTF) to follow up. I'm still very skeptical that this all originated with google searches, but the story of "JTTF agents" checks out.
So it's certainly possible that three black SUVs worth of the local sheriff's deputies, all of whom were members of the Joint Terrorism Task Force, showed up to question her about her internet searching.
Or this fiction writer is not portraying an accurate version of the events.
The officers are basically deputized as JTTF members. It's a common model that's been around for a long time. For example, the US Marshalls have a fugitive task force for picking up suspects with arrest warrants and the DEA has something similar for drug raids.
So FBI became JTTF, which in turn became Nassau cops?
This is why we should all stop listening to the 24 hour news machine and wait a while before everything is cleared up and fully presented. This is a developing story and clinging to one or two tidbits finagled from the authorities here and there aren't going to cut it.