There probably isn't a specific permit or process for two primary reasons.
Firstly, crashing a plane intentionally is a rare thing and (until now, apparently) isn't the kind of thing the riff raff do so there's no reason to throw up barriers to keep the riff raff out nor enough volume requests to warrant a an institutional process for dishing out said denials.
Second, it is naive to expect a government agency to stick their neck out and accept some degree of liability for something like this (which is what approving a comprehensive plan would do). It's not like there's highly developed industry standards for plane crashing they can rely on.
The closest you'll get to a "permit" is probably the FAA knowing your intentions in advance and not objecting.
I assume there's some permitting regime, and of course you'd need to find a place to do it, but probably nothing too onerous by the standard of being able to buy a whole airplane specifically to wreck it.
edit: I was close! It's a Boeing 720, a short-range variant of the 707. https://youtu.be/Jpc32JQXT_0