While it may be possible to get it into an 82" door, it is going to be a stretch to say the least! And a ton of work, but if you can manage it more power to ya! Just to give you some perspective, I do not have a VF3 or 4. I have a 2002 VF2SS. All leveled up and ready to run, the top of the electrical cabinet and arm that holds the pendant is at 76" from the floor. It is 86" to the top of the regen unit, (coffee pot warmer), and about the same to the top of the sheet metal enclosure the track mounts to. When we moved from our last building, they picked the mill up with their forklift, raised the leveling screws up into the casting, and leaving the forks dragging on the floor, the side mount tool changer cleared the 93" beam down the middle of our building by no more than an 1/8". I also had to take the track loose from the head and electrical cabinet and hold it down to clear that same 93" beam.
Now, having said that, you are looking at an umbrella style tool changer which should take that completely out of the mix because, as stated above, that should be inside the machine and not above it, or at least not far above it. Also, for that year, the SS head was somewhere between 8 and 10 inches taller than the non-SS models. So that may just fit...BUT...The VF3 and 4 also adds another 5" of vertical travel that my VF2SS does not have. So if we were to assume the cabinet is 5" taller, it's going to be close! I guess, worse comes to worse, get the sawzall out and make the door a little bigger!
Good luck to you and let us know how it goes!