If I took a Tesla coil into space, and then turned it on near, but not touching metal objects- would charge jump to the metal objects? It's my understanding I wouldn't see a flash, since the lightning comes from ionizing the air, but would the charges still move?
The conventional definition of lightning is a current though a plasma (not necessarily through air as lightning happens on other planets) so in a vacuum there cannot be any lightning. However charge does still flow between electrodes in a vacuum, and from personal experience I know that we can get something very like lightning in the right circumstances.
In a vacuum charge can leave a surface due to field emission. Basically the field gets so strong that the electrons at the surface can get enough energy to overcome the metal work function by moving only a small distance and they tunnel out of the metal. However this is a rather diffuse flow and wouldn't generate any light so it wouldn't resemble lightning.
I did claim you can get an effect like lightning, and this can happen if there is a rough spot on the electrode surface that concentrates the electron flow. If the electron flow gets concentrated enough it can vaporise the electrode at the high point then you get an arc through the vapour. In the old days when we used giant Van de Graaff generators as particle accelerators this used to happen on a regular basis leaving a scar on the dome that then needed to be polished away.