Based on my answer to the duplicate question you need to do 7 levels of damage to incapacitate a Vampire with a single shot.
With the 2nd edition and 20th anniversary rules it's pretty hard to do more than about 3 levels of damage at once unless you manage a fabulous roll.
So you've got to find 4 extra levels of damage from somewhere which equates to about nine or so extra dice with 'average' rolls.
The Barret 50 rifle and other similar 'anti-material rifles' are the biggest guns you can get that are remotely sensible in game but the only listing I found for one was in the 1st edition Hunters Hunted supplement where it does 4d (I assume its 4 extra dice). In the real world they can fire a variety of incendiary, high explosive, incendiary high explosive, and High Explosive Armour Piercing Incendiary rounds. This sort of round
is a suitable round for engaging helicopters, aircraft and lightly
so it should be more than a match for any Vampire's skull.
However there are no written rules for any of these rounds that I'm aware of so you're going to have to persuade your game master to either add some rules or allow you to role-play a situation where the outcome is a single shot kill.
Frankly its almost certainly more fun to do this via story telling than trying to write some sensible rules that cover these sorts of weapons without going overboard.
You'll need to stalk your target, work out their routines, hideouts and lairs, find a place to shoot from, wait patiently for the right moment and then finally take the shot. All of that is much more dramatic than simply being given a 'Big Vampire Killin' Gun' from your sire and blasting away.
For the mechanics of the kill shot I would rule that you need to effectively decapitate the target. i.e. Don't worry about the levels of damage or the damage type just work out the effect.
I'd start with a given that most creatures (including humans) can survive a rifle shot to the head (i.e. one that does 3 or so levels of damage) by ruling that this sort of shot does not penetrate the skull or damage the brain or spinal cord.
Thus what we need to allow for are shots that damage the vital areas of the head resulting in a kill on humans and a knockout on Vampires and Garou. at the very least this needs to be an Aimed Head Shot at difficulty 8 but it might be betterto make it a special attack with difficulty 9 but with any success resulting in a kill. There's a big difference in terms of chance to succeed between difficulty 8 and 9, see this question for details, so be careful about imposing this higher difficulty. It might be better to use a lower difficulty but require more successes.
I'd also say that in order for this to be successful then only certain weapon and ammunition combinations can accomplish this sort of kill against a supernatural target. Against ordinary humans and Ghouls then any sort of moderate to high calibre rifle will suffice but for Vampires and Garou it needs to be something more such as the anti-material rifles above coupled with high explosive and/or incendiary ammunition. Although an 'ordinary' sniper rifle with such ammunition might suffice depending on your needs.
In many ways though the precise weapon and ammunition combination doesn't essentially matter, as it's a flavour component, what matters is that such a weapon is not generally available and it's use is restricted by 'the elders' to prevent an escalation of weapons and/or to keep to the masquerade. If you espouse that sort of restriction in your campaign it makes their use much more playable than otherwise.
Thus the first thing your player will need to do is to ask for permission to assassinate their target, this in itself will make for an interesting session as the player needs to come up with a valid reason (or excuse) to use the 'forbidden' weapons.
I think that covers most of your points but to answer your point about safety - this sort of ammunition is safe to use (for the person with the weapon anyway) if it was dangerous just to carry around then it couldn't by used by real life military personnel, even considering their much higher accepted risk level than civilians.