I would like to know if Video Kiosk will work with the Amazon Fire OS based Fire TV sticks.
The short answer is no, if you want to run Video Kiosk you should choose one of the many other Android devices that are available on the market.
The long answer is, Amazon heavily customize the Android OS, so much so that they call it Fire OS. While Fire OS can run Android Apps (so technically Video Kiosk will run on Fire TV boxes) there are so many ways that Amazon limit their products that it makes them unsuitable for use in kiosk operation.
For example on Fire OS it is not possible to disable the Fire TV launcher, it is not possible to provision Video Kiosk as a device owner, it is not possible to add any Account except for an Amazon account, it is not possible to use the Google Play store, it is extremely difficult to root the device, and the list goes on.
The bottom line is that Fire TV devices are designed by Amazon to sell Amazon products and services to such a degree that I don’t recommend anyone purchase them for any other purpose.