Can I put my NSF fellowship on reserve in order to investigate leaving academia? I'm currently a PhD student in mathematics at a well-ranked school. Unfortunately, after a few years of increasing frustration and self-doubt, I'm considering leaving academia and trying to start a different career.

I currently have an NSF graduate fellowship. As described in the information guide for fellows, one's "status" with regards to the fellowship can be either


Tenure: Tenure is the period of time during which Fellows utilize
the fellowship to pursue an advanced degree in science,
mathematics, or engineering...
Reserve: With prior Foundation approval, Fellows may reserve any
two years over the five-year period to use an alternative means
of support, or to engage in activities other than graduate study... (emphasis mine)


(There are also options for medical or military deferral. The full description of Tenure and Reserve status are in the linked document.)

Does this mean I can request to reserve my fellowship for the 2015-2016 academic year, using that time "test the waters" on another career, such that if I decide to stay in – or rather, go back to – academia, I would still have my NSF fellowship? 

Of course if I do something like this, I would want it to be with my advisor's approval, though I haven't brought up this specific idea with them yet. Let's assume they are okay with it.
There do not appear to be any reservations on the purpose for requesting a deferral in the linked guide. You need to tell them in advance of filing the "Declaration To Utilize, Reserve Or Terminate A Subsequent Year Of A 3-Year Graduate Fellowship Award" and get their approval, but I don't think that there are many reasons for them to deny it. You might want to have something lined up like an internship, other job, or visit to a foreign university rather than just saying that you're taking a year away from studying. Though, I wouldn't be surprised if the latter happens, too.

Edited to add: You need to start talking to your supervisor and eventually the NSF Program Officer in charge of your fellowship.