The VR-4 had the 4G63 engine, similar to the ones in the Eagle Talon, Mitsubishi Eclipse, and Plymouth Laser (as well as Lancer Evolutions). All of which, minus the Evo & VR-4, had FWD turbocharged versions. The VR-4 was also only sold here in the U.S. from '91-'92, so obviously it's a completely different chassis as the Galant that you have now. Of course there were also other "newer" AWD Galant models in other countries, with twin turbo V6 engines. So depending on how different the US chassis are to the JDM or EDM chassis, it might not be too huge of an issue to install a RWD or AWD drivetrain in there. Of course, it will still involve buying a ton of parts from other countries, since your car is set up for fwd right now.
However if you really want to keep it FWD, you're far better off finding a good engine & transmission that is already set up for FWD. There are plenty of FWD cars that are factory turbocharged, or can be build to handle forced induction. There's no way your existing transmission will bolt right up to a Nissan engine. If you're really lucky, you might be able to find a company to make you an adapter kit, to mate them together. You'll be spending quite a good chunk of money for it though.
But if this is your daily driver, you should consider buying a new one while you work on your current car, not after the work is done. Even with a FWD engine/transmission swap, you're still going to do a TON of wiring changes, mount fabrication, axle fabrication, shiftlinkage fabrication, etc. Unless the swap you choose is made specifically for your chassis, your car is going to be out of commission for quite a while.
Not to be rude, but if you have to ask if your transmission will bolt right on to an engine from a completely different manufacture, then you're in over your head already. I think it's safe to say that most car enthusiasts dream about taking a light chassis and doing some kind of unique swap in it. 95% of the time it's just a pipe dream, unless you have some serious fabrication skills & knowledge, or know someone who does & have deep pockets.