Not with the internal flash. In continuous shutter mode, the internal flash will fire only once each time that you press the shutter button. Most of the Nikon manuals specify those words, but I don't find it in the D5200 manual. For example, page 148 of the D7200 manual, or page 186 of D800 manual.

Speedlight flashes must recycle to fire again correctly, however, most hot shoe flashes can recycle that fast if at low power level. But hot shoe flashes will try, regardless if they are able to recycle or not (has to be low power to work).

But the internal flash is different, handled more carefully.

Not with the internal flash.

In continuous shutter mode, the internal flash will fire only once each time that you press the shutter button. Most of the Nikon manuals specify those exact words, but I am surprised that I don't find it in the D5200 manual. For example, page 148 of the D7200 manual, or page 186 of D800 manual.

Speedlight flashes must recycle to fire again correctly, however, most hot shoe flashes can recycle that fast if at low power level. But hot shoe flashes will try, regardless if they are able to recycle or not (has to be low power to work).

But the internal flash is different, handled more carefully.