So here's the deal: I read that using Paralyzing Palm to stun NPCs would allow you to go through their inventory and take whatever you want with no repercussions. However, whenever I knock someone down, the only thing that pops up when I try to access their inventory is: "{NPC Name} is unconscious!" I know this worked in Fallout 3. Does it not work in New Vegas? Or is it a bug with my game?

They may have changed that, they made a quite a few changes to make it more challenging, but one of the key things was being able to complete the game without killing anyone. this would be difficult, as you must knock out opponents occasionally to get by a tricky situation. if this were the case, I can't see why you wouldn't be able to loot the body. have you tried going into sneak mode and pick-pocketing their items? maybe because their still considered alive you can't "search" or loot the body.