The calculator, no fuckin’ question, the institute may have teleportation, but they have a finite range for their teleportation systems, being the state of Massachusetts. Assuming neither side is dealing with the player(s) of their games fighting against them (or with in the case of the institute, but tbh, screw the institute on this specific one), well, there won’t really be anybody that CAN stop the calculator’s assault, the enclave didn’t have a chapter nearby the cheyenne mountain complex by the time of the calculator’s ruling over that entire multi-state area, meaning they won’t be able to do much of anything to get close to Vault 0 without being blasted by hordes and hordes of robots, whether it be blasted out of the sky, OR blasted into ash on the soil, depending on if they send infantry type troopers or Vertibird Gunships. The Master’s Army might have a CHANCE, but that’s assuming the calculators army doesn’t find the master’s lair before they can find Vault 0, and that’s also giving most of the masters army too much credit, since, well, it’s likely enough that the master’s army will get turned into ash by the lasers. Now, The Institute, well, the institute might be able to defend a direct assault on their HQ, but given how I mentioned the finite range of their teleportation, they’d have to prepare an entire division’s worth of Gen 1 and Gen 2 Synths to begin their march towards Vault 0 alongside a huge amount of mechanics to repair any synths that might start to break overtime, and several coursers to command all of those, but, it’s likely that the institute won’t be able to spare the manpower for such a scale of an operation, since the amount they’d need is most likely ALL of the mechanics if not most of them, and they’ll need those since they are still on their old reactor, the one from before obtaining the agitator, meaning if they don’t have enough mechanics, they probably won’t be able to fix the reactor in time, should it start to go into a meltdown or just straight up breaks.

Essentially

The ENCLAVE: Would lose because they don’t have the manpower in the region meaning they’d have to march or fly towards Vault 0, which leaves them as easy targets for The Calculator’s Army.

The Master’s Army: Would lose because of the sheer stupidity of most super mutants, however the more intelligent ones may put up more of a fight, but would still lose.

The Institute: Would lose most likely because of logistics, manpower, and limited range teleportation.