I think this would work generically for any type of Sorcerer, maybe I will make a subclass for it but it's not needed.  If you think of each level spell slot as a quality of ammunition for a gun, and they "cast" a spell by using a round of ammunition (a spell slot).  The character can only make more ammunition during a long rest or by using sorcery points (just like spell slots).  The ammunition are generic and can hold whatever spell the character knows. You literally don't need to change anything, you just flavor everything as firing a gun instead of casting it.  The interesting thing is that, in a real way it makes spell slots make way more sense.  Instead of being made up limits they are actually physical things in the world that it would make sense to run out of.  If you have a gun, your going to run out of ammunition at some point.  I would make it that all the rounds are in a selective magazine so there's not need for reloading.  When your clip is out, your done anyway.

That's some of the flavoring attached to the Artificer in UA... where instead of having spells the character uses science to essentially recreate the effects of spells. If this is just for flavor, though, I think a good DM should allow it. Functionally its identical to normal casting anyway, but if it lets a player have more fun and interesting roleplay opportunities then I'm all for it.

Would it make more sense to make this a wizard?  At first, I saw your gun as a reskinned arcane focus, but the sorcerer would still have access to Verbal and Somatic spells. If you imagine the gun as a spellbook instead, then the wizard would be in a more disadvantageous position if the gun is missing. Preparation makes a little more sense since you're "preparing bullets."

You'd also have to remember to only take spells with a range (no touch or self spells).

But as stated above, the flavor of the Artificer is that they're throwing/launching vials/contraptions/whatnot instead of casting spells.

I was thinking of touch spells as shot gun shells.  It's not that they only effect a really short distance, but they are only effective for very short distances.   

One thing I did forget about was the whole noise thing.  Instead of just having to say some words now you have a huge boom every time you cast a spell.  If we're in a populated dungeon, all the monsters know your there.

P.S I picked sorcerer simply because I feel they already have much more thematic / restrictive spell casting.  It would make more since for a Sorcerer to only spells that make sense as ammunition (There are spells that wouldn't work, booming blade comes to mind.  Your already using a gun, you can't melee at the same time.).   The elemental sorcerer is a thing, so I can use that.  I feel as a DM I could control the spells they have access to more and not have it be unfair.  Wizards, by definition, can cast anything.  They don't have that built in limitation on spells.    You pick a Wizard to be able to cast everything, I don't want to get in the way of that.  Just my thinking.

Well, I think it would make sense to find alternatives to the gun for any spell that wouldn't make sense as a gun. Like... casting invisibility on yourself might take the form of a small smoke bomb that covers you (or someone else) in enchanted smoke that grants invisibility. Something like Shocking Grasp could simply be a taser glove, or even booming blade could be explained as a sonic device attached to your sword. Although I suppose at that point the focus is shifting from being a Gunslinger and into being more of a general inventor/tinkerer...

You could just have invisibility act as a grenade effect of 5 ft you shoot down.  Or we could go all persona 3 and have you shot yourself in the head (a RPG where you shot yourself in the head to summon spirts to aid you, yes it was somewhat controversial at the time).

The slapdash way to go is basically the gun is a wand that your player NEEDS in order to cast spells. That and the fact it's loud is two disadvantages with no upside, however.

Suppose you make it a sorcerer subclass? The gun could then have gun-like metamagic feats, like cone effects (imagine a shotgun Lightning Bolt), plus you only have to think of stuff at various levels instead of reimagining the whole class.

Just realized some interesting things about making spells slots real, tangible things.  They're physical things that can now be traded, or looted.  I would say two sorcerers could trade ammunition.  It would cost a action from the sorcerer giving the ammunition and a bonus action to load the new ammunition by the sorcerer receiving it.  I'd also say they could take ammunition from a downed ally for both an action, to find it, and a bonus action, to load it.  However you can only load up to your maximum spell slots for any level and only up the levels you have.  For example if you have used 2 of 4 1st level spell slots and you get 4 rounds of 1st level ammunition you can load 2 ammunition, you need to leave the other 2 behind.  If you find a 3rd level ammunition and you don't have access to any 3rd level spell slots you can't use it to get a 3rd level spell slots.  I'd even say you could share you ammunition with the other casters in the party to give your spell slots to them.  This could be cool if your a fire sorcerer and the enemy is immune to fire, you can share your spell slots with another caster that can cast other elements.  Your basically in a support role now as a supply of spell slots.  I don't feel like this is over powered because the group still has the same amount the spell slots over all, it's just that now you can share them between the party.  If the party can cast 3 fire balls, they can still only cast 3 fire balls, only now one person can cast all of them where maybe before it would take two or three.  It's just an interesting Idea.  Still thinking on if it would be possible to get ammunition from enemies.  Not sure on that, unless you want a real high magic game, then yea, why not.  That just means more magic.  You can give the party more or less ammunition as you see fit without needing a long rest.  I kind of like that because then you can replenish only there spell slots but not there health and other abilities that reset on long rest.  It's interesting.

PS I got this Idea from fairy Tail the anime.  They have two characters that use guns as there magic.  One of them looks like a desperado type, the other is a sniper.  

At this point I agree it's getting to be more than just a reskin for the sorcerer and more looks like an entire subclass unto itself (if not a completely new class)

Having ammunition, but still being limited by your spells per day sound more arbitrary than your initial argument.

If you look at your ammunition in a different way, they're reskinned spell scrolls. Also from what you're describing, you probably shouldn't constrain yourself to reskinning the sorcerer.

It sounds like you could create a new class (doesn't even need to be a spell casting class). This class creates some number of "ammunition" after each long rest. That number probably increases with level, or instead the maximum number of ammunition is dictated by level and you create some stat bonus number of ammo, like Int Mod (min 1). 

The ammo only works when fired by some "gun" capable of loading said ammo (that doesn't have to be class dependant).

At various levels you learn ammo effects, like a fire or lightning bullet. You also sound like you have two subclasses, a pistoleer and a sniper, who probably learn gunplay things, like the pistoleer learns cone effects and the sniper learns line effects.

They make ammunition by summoning it.  There's no cost, or time involved.  Were not talking about two limits.  Only the one.  I should have made that clearer.  There is only spell slots.  I'm just making those slots a thing as opposed to something made up.