Can I connect a DC high voltage booster directly to my Raspberry Pi?
The advert says it needs 5 Amperes. That’s hundreds of times more than a Raspberry Pi GPIO can supply.

You would need a driver, preferably with isolation, and a separate power supply capable of at least 5A. One solution would be a CPC709J, with a suitable series resistor for the LED. 

You may well cause disruption or damage even with that, but the chances are better. You’re essentially putting  a multi-watt spark-gap transmitter in close proximity to an unshielded microcomputer.