Well..... first, what terrains would your VX be traveling upon?
If it is a street queen, the stock axles should hold up fine IMO. You may be able to retain TOD that way, simplifing things. You would have to investigate the spline count for the input shaft of the t-case as well as the bolt pattern of the mount to the trans to discover what options you have.
Don't limit yourself to just an LS1... let's say LSx. So then, if you are serious about the swap and figure out you can keep your t-case, be on the lookout for an engine+trans combo with computers. Otherwise, you need to find also a t-case with a passenger side drop that can bolt up to your new trans. At that point you may or may not have to change out your driveshafts. With a swap like this, modification to the rad support, firewall, trans tunnel, and perhaps floor may have to be performed to allow fitment. LS truck motors are dimensionally the same as the cars but the belt-driven accessories, intakes, and oil pans are different. So be on the lookout for car motors versus the truck motors to simplify fitment. Locations of the A/C compressor vary (low mount, near frame rail, would be problematic) and there are solutions to relocate - some affordable, some around $2000, believe it or not. I got my used complete 2006 aluminum 5.3 LS with 4L60E trans and wiring harness for $2000. Steering shaft, exhaust manifolds/headers, heater box, A/C system, radiator, crossmember, front diff, driveshaft, starter, oil pan, hood, inner fenders, etc... these are all potential clearance issues to address.
..............Good luck...............
Sent from my "two hands on a keyboard"