Putting the volcano in the vacuum would cool it much less effectively than leaving it in the atmosphere. An atmosphere at least provides a little bit of mass to carry away some energy by convection – but in the vacuum, there is nothing to carry heat except radiation.
Vacuum is actually a very good insulator. Remove the atmosphere and you’ve essentially put the volcano in a thermos.
By the time you get to the entire ocean and it’s still not enough, I think it’s safe to just go ahead and say “no amount”.
Except that it’s false that no amount of water is enough. The problem with the ocean is that it’s just sitting there. You aren’t moving the water onto the heated surface of the earth and then off of it quickly enough.
How about the problem that blocking the vents merely forces the underlying pressures to build up further, running the risk of an explosive eruptive break through?
Human action relative to tectonic forces is trivial. Might was well ask how ants could work together to block the Danube.
IANAERKATS (I Am Not Anyone Even Remotely Knowledgeable About This Subject) but generally speaking I think it is desirable to keep the lava underground. If you don’t allow it to break surface it will hopefully continue on its merry way deep into the bowels of the earth where high pressure is is the norm.
Mate, it’s coming up because pressure is pushing it up through cracks / weak spots / separation points.
It is not “flowing down”
So your hope is deeply, badly misplaced.
Just where do you think it came from in the first place?
As people keep try to point out, this is impossible.
A volcano is literally caused by the earth’s heat melting part of the mantle or crust. You can’t turn off the earth’s heat. You can’t keep the pressure from breaking through by stopping it up at any one point. You can’t cover the entire earth with an impregnable barrier.
If you poured all the oceans into the interior you would boil them away but not cool the earth. The earth is 8000 miles in diameter, the oceans are a tiny film. The earth is thousands of degrees warmer as well. We’re lucky that the crust is as good an insulator as it it, but when the internal heat melts a spot it cannot handle the pressure. It can’t go down, because that is under much higher pressure. It can only go up. If several miles of rock can’t stop it, then nothing we can come up with can do so.
You’re wrong because your basic assumptions are wrong. Try reversing them, then you’ll see what the others have said snap into sense.
This volcano is already under a freakin’ glacier. Can’t do much more cooling than that.
You forgot to include a cite for the point that “you can’t keep the pressure from breaking through by stopping it up,” and you accidentally included snark at the end of your post. Oops, me too.
Nope. That’s I wrote after removing the snark that was in the original. 
We’d be better off building a big-ass tube over the volcano cone so that when it erupts, it shoots all the particles upward and into space high enough to avoid darkening the sun.
An explosive volcano consists of the pressure blowing a stopper made of several kilometres of solid rock into the stratosphere, AFTER the pressure has already forced it’s way through the entire thickness of the earths crust.
If natural processes can’t keep the pressure from breaking through despite sitting e.g. Mt Pinatubo or Mt St Helens on top of the weak spot, what more cite do you need?
So you want to construct a supersonic pressure washer that can carry as much flow as the amazon and train it on the volcano? I suppose that might be able to carry away enough heat to make a difference, but would probably cause more problems than it would solve - for a start it would probably erode a big hole in the crust, directly above a huge magma upwelling. Ooops!
Wait, what? Did I miss the news? Are we all about to be killed by a volcano?
Can you be more specific about what you mean by “about to be”? In my case, the increased consumption of beer caused by time spent beach-side while waiting to see if I can fly home to Canada because of the shutdown of European airspace due to a volcano might lead to my eventual death. I suppose that’s “about to be” on the Earth’s time scale…! 
The cite would be to bloody Geological sciences. Some things are just a wee bit too obvious and basic to actually provide a direct cite to, as no bloody fool actually would waste time on a direct study of it.
Volcanism is a geologic process, and that magma is pushing through kilometres of dense crustal rock. The forces involved are … planetary. Freezing a cap only allows more pressure to build up. This is known, in fact, it’s a known factor in explosive eruptions from long dormant craters. Things like Krakatoa eruptions are not good.
It’s far, far better that the magma reach surface and relatively peacefully release built up pressures.
Rather than simply arguing with everyone with a clue that you can’t accept the facts as such, perhaps some own reading is in order.
Suppose, for the sake of discussion, we need to solidify a plug of lava 100 meters wide and 20 kilometers tall. Typical temperature is 1000C, suppose we need to cool it to 500C to render it firm enough to prevent further upwelling.
Granite: density 2700 kg/m^3, specific heat 1.017 kJ/(kg*k) (these are for solid granite, I’m going to assume they’re the same for lava/magma)
Based on these numbers, we need to remove 216,000 GJ of energy.
Water: density 1 kg/m^3, liquid specific heat 4.184 kJ/(kgK), heat of fusion 2257 kJ/kg, vapor specific heat 1.872 kJ/(kgK). So to take 1 kg of water from 20C to an average of 750C requires 3809 kJ.
We need 216,000 GJ/3809 kJ = 56.7M kg of water, or 56,700 cubic meters. That’s a cube of water 38 meters on a side.
That’s a manageable quantity of water, but as has been noted, the challenge is getting it where it needs to be. When the Hoover Dam was made, cooling coils were built into it so they could circulate chilled water and remove the heat of the curing concrete. No such system would be possible for cooling a 20-kilometer-tall plug of liquid rock.
For some perspective, check out the lava cooling operation effected during the eruption of Eldfell in 1973. Initially they were pumping 0.4 cubic meters of seawater per second onto the lava; that means every 40 hours, they dispensed our 57K cubic meters of water. This went on for nearly two months. By the end they had brought in a seawater pumping capacity of ~32 cubic meters per second. That means every half-hour, they dumped our 57K cubic meters of water on the advancing lava flow. Not clear how long they employed this much larger pumping capacity, but it finally worked.
saw mention where because of the glacier there is more cooling producing more ash plumes and explosions.
I’ll assume your claim of 130 short tons of liquid helium is accurate.
Helium heat of vaporization is 21 kJ/kg.
vapor specific heat is 5.1926 kJ/(kg*K).
Assume the liquid helium is already at its boiling point (~4K).
To take boiling helium up to an average temp of 750C (1023K) requires 5333 kJ/kg.
with 130 short tons (118,000 kg) of liquid helium, we could remove a total of 629,000 MJ of energy from the lava. This falls short of the requirement in my previous post (216,000 GJ) by a factor of ~340,000.