It is highly unlikely that you would be able to draw power out of the charging port. Simplistic devices sometimes let you do that, but there's probably a charging circuit in the laptop which would be in the way.
There may be an internal supply of greater than 12 volts which you could tap into by modifying the laptop, but that's far from certain (not much today would need that voltage - the display backlight would need more, but that probably doesn't have capacity to spare).
The most realistic path would probably be to use one (or more?) USB connector(s) to feed a DC/DC converter, provided that the power your require divided by the converter efficiency is within the USB wattage spec (which thinking simplistically would be a little under 2.5W after negotiation for a normal port, maybe 50% more for some special ones intended for charging phones)
You could also examine the video receiver and see what it really requires - there may be a 3-terminal regulator on there which you could bypass (if you were very lucky it would be a 7805, ie requiring 5 volts downstream). Though RF stuff, at least if it transmits, sometimes does need the higher voltages.
The quality of your power supply - if it is contaminated with digital noise - may degrade your video signal (or even cause spurious emission outside of the intended frequency range).