Not natively that I know of, but the Advanced Scene Switcher plugin can at least *start* a fullscreen projector:
This plugin will allow you to automate various tasks using "Macros". Macros consist of a list of conditions under which a list of actions will be performed. Examples and guides can be found in the wiki. Feel free to contribute! If you run...
obsproject.com
If you still want a hotkey to do it, you can create a condition for that, but if the hotkey is really a concession for something else, you might just find the something else in there too.
The difficulty comes in closing the projector, because it's its own separate window that nothing seems to have a record of, according to the author of that plugin. Seems like it's just barely attached to the rest of OBS, just enough to get its signal and no more. So it's hard to grab a hold of again and do something else with, like closing it.
That also means it's practically impossible to tell if the one you want to start is already started. And there's nothing preventing you from having multiples of it. So be careful.
