The flashes used for surveillance photography during WWII were basically large flashbulbs, burning magnesium, but not what Edgerton used for that photo, it's stated in the description he used an air gap flash, which, while bright, mainly serves to have short duration, not extreme brightness over distance.
Wikipedia has a good description of what he used for most of the really high speed photography, an air-gap flash.
https://en.wikipedia.org/wiki/Air-gap_flash. It's very similar in principal to the normal Xenon strobes we've all used, but usually uses nitrogen, in order to get a much shorter flash duration. They're not really 'off the shelf' but there are plans out there. Doing some searching around, there is a company here:
https://shop.vela.io/products/vela-one-high-speed-photo-camera-flash that claims to have made an LED flash capable to the short duration required for capturing extremely high speed motion. All that being said, half the battle is the lighting, but the other half is timing everything to work as desired. Fortunately, with the rise of more affordable high speed video cameras, triggers for just this sort of thing are pretty widely available.