I play snipers and here is my build at level 22:
S 2
P 4
E 4
C 6
I 6 (+2 since rolled)
A 5
L 1
Sneak 3
Lone Wanderer 2
Gun Nut 2
Rifleman 3
Locksmith 3
Scrapper 1
Local Leader 2
Medic 1
Cap Collector 1
I'd say the most important skill for a sniper is sneak. Gun Nut is probably second so you can trick out your rifles how you like. Rifleman of course is very helpful; But even without any Rifleman perks, a well placed head shot with a strong enough rifle should either kill or knock to half health most run-of-the-mill baddies (not bosses). Of course the goal is to one shot them and not have to deal with them shooting back or charging at you, but if you knock them down by half health, it is still good.
The trick with playing a sniper is to be very patient. Observe, move slowly, wait, get in position for the head shot, wait, wait, wait . . . finally the perfect head shot and *bang* dead Super Mutant/Feral Ghoul Reaver/insert baddie. The game engine does a good job of "weighting" anatomical hit values, namely headshots are devastating. The fact that the sneak attack multiplies it by 2 or 3 times is of course not realistic, but it is a satisfying and not unbelievable game dynamics. It is generally advisable to shoot once and relocate, or at least fall back to where enemies will not detect you.
There is no point playing a "sniper" who uses VATs. It is just not the way it was meant to work.
A VATs toon would tend to use pistols, sub-machine guns, maybe shotguns or carbines, melee could be good for a VATs build too, though the need to invest in Agility, Strength and possibly Perception might be a bit of a trick.
ADDIT: I read your post a bit more closely and I think you are still not clear on one thing, you would not use the same weapon in VATs as you use to snipe with. My load out is like this (by hotkey)
1. "Big Jim" a Legendary hooked pipe wrench that has a chance to cripple legs
2. "Sgt Schultz" a fully tricked out combat shotgun that does about 80 damage per shot
3. "Spring XdM" a tricked out silenced 10mm pistol, non-auto. I'll upgrade to a more potent pistol at some point.
If I ever use VATs this is what I use, though the shotgun could be effective too.
4. ".38 Lee Enfield" a pretty heavily tricked out .38 calibar pipe rifle with glow sights. Reflex would be fine too. This is my "pea shooter" that is for killing things that get too close but are not too threatening.
5. ".308 Nightstalker" a heavily tricked out pipe rifle chambered in .308 and with a night vision scope. I pretty much don't bother changing to a day scope at this point, it is doable. At one point I had a nice Recon sniper rifle that was sweet, but I guess I accidentally sold it.
Between Sgt Schultz and Nightstalker that is probably 80% of my kills, and Big Jim is probably another 10% (use it on mongrels and mole rats and the like).