Ok, so first off, your vacuum isn't really a vacuum, or else nothing would happen unless you were able to force electrons to jump the gap on their own, leaping from one metal to the other. Your "plasma" is created by ionizing the gas within the chamber. You can create plasma at atmospheric pressure as well. Lightning does it all the time. What you're really asking is why it's easier to create a plasma (ionized gas) at low pressures than it is at higher pressures. That's pretty easy to explain in terms of mean free path (MFP) of the charged particles within the ionized gas and the dielectric screening effects of the gas itself. To get the gas to conduct, you have to knock electrons off and get the electrons and/or negatively charged ions to move opposite the positively charged ones, thereby becoming a conductor. But when an ion hits another oppositely charged ion, poof, no charged particles and no desire for them to move in a given direction. That's the MFP part. The vacuum increases MFP because the atoms are further apart. (Of course if we're talking air here as opposed to a noble gas like neon or argon then you have to think about the molecular nature as well.) So then there's the dielectric nature of the atoms/molecules themselves, which, when exposed to an electric field, will polarize to screen that field. Positive charges will go to one end and negative charges to the other and thus the net localized field is reduced. The more atoms you have in a space, the higher the dielectric and the stronger the screening effect. Thus, the higher field it requires to get the charges far enough apart to completely dissociate. The same is generally true of any dielectric insulator, not just air.

So, yes, if you lowered the pressure outside the tube to the same as inside, you'd get arcing outside as well.

The one other piece of the puzzle here is the frequency of your signal, which allows the plasma arc to form inside a completely insulated tube. If you were to apply DC of the same voltage, the charges might split and go to opposite ends of the chamber, but then they'd stay there and there'd be no visible current flow or plasma arc. The peak voltage still has to be enough to ionize the gas, but then current flow alternates, essentially bouncing the charges back and forth over some small distance.

Hope that helps.

As Michael D. Foegelle says, if it was complete vacuum nothing would happen. Breakdown voltage drops as the pressure drops, until very high vacuum, when breakdown voltage increases again. That is why high-altitude radar systems are pressurised, to prevent breakdown caused by high power microwaves.

Malcolm White reminded me of another application. At the start of my professional career, I was hired to work on a lightning/EMP generator product line for MIL-STD and DO-160 testing. These all used spark gaps as triggers for extremely rapid high voltage charge release. Basically you charge up a big capacitor or two with the voltage applied across this spark gap. The spark gap has three terminals, two at the ends and one near one of the ends. You apply a pulse to the third terminal which causes an arc across the small gap, and once the gas inside is ionized, now the spark jumps across the larger gap, releasing the big charge into whatever pulse shaping circuitry you have after that.

The reason I bring that up is that the largest unit, a 100 kV exponential pulse generator, had basically a metal bucket with an oil bath that had two big 50 kV capacitors on either side of a large spark gap housing (still have one as a decoration in my office). The capacitors were charged in parallel with a resistor ladder and then fired in series through the spark gap, doubling the voltage they were charged to. What was unusual about this though was that the spark gap had a couple of air fittings on it, and you varied the pressure in the spark gap to control the breakdown voltage. The goal was to have the pressure JUST above the breakdown voltage for the gap, so that it wouldn't self discharge, but would break down rapidly when triggered in order to have the fastest switching time possible. So tuning the output level of the pulse was both an electrical voltage setting on the power supply and a pressure setting in the spark gap!

Thank you Michael for clarification and detailed answer.