I know this is a bit odd, but I think the game could potentially have a place for old fashioned guns in it.
The reason for this is that modern firearms require ammunition that can’t be produced ingame, and would be a bit overpowered if it could be produced in any sort of volume. This makes specializing in them a bit… unwise? As it’s not a sustainable strategy.
To counter this I propose that the game include simpler firearms. You could make them out of scavenged parts, so you wouldn’t necessarily have to make the mechanisms and fine components yourself, but the idea is the ammunition should be simpler. Rather than needing primers and hulls, you just need to melt down the lead (possibly also producible from scrap metal) and create some low grade black powder to load it with. Oh and maybe some paper/rags to make up some pre-packaged cartridges. Soldiers did make their own ammunition in the time of musket weaponry, so this definitely fits the survival situation, i think. Black powder could be either made by cutting regular gunpowder with something else to bulk it out, or possibly created using a good bit of cooking skill from other chemicals found around the game world. Either way, black powder should be much more abundant than regular gunpowder.
You could potentially make both muskets and rifles, muskets being very powerful, wide bore weapons that are inaccurate as hell, and use ball rounds. Rifles conversely are much more accurate but you could require them to use a salvaged barrel from an existing rifle, which would be a good bit smaller than a musket barrel and thus the rounds would be less damaging, but the gun would be useable at longer ranges. Rifles should also use bullet shaped ammo which would have better armor piercing.
Naturally being muzzle loading, these would all be slow to reload, single shot weapons, as a tradeoff for their sustainability, and you could make a variety of variants of them too, such as Old Fashioned types, which are big and heavy, but make good clubs and can be made partially from wood (these could also be found in mansions and rarely mounted in bars). Salvaged variants, made of modern parts and thus much lighter, but less good as a melee weapon, and requiring more metal, tools, and mechanical skill to make, a Pump Action version might be possible? Not sure if you could make something that could load paper cartridges using a pump action, but it would work a bit like a shotgun, though being a homemade gun the recoil would make it hard to fire it rapidly and it would not be compatible with some of the possible mods.
You could also have some variations which can be done as mods, such as Breech-loading/break action variants, which are faster to reload but lose a bit of power due to an imperfect seal around the action, or perhaps they can only fire the lowest power cartridges. Double Barreled variants, which can load and fire two barrels at once, or sequentially, but which are heavier and less accurate than their single barrel counterparts. Reinforced variants, which gain extra melee damage and can fire overcharged rounds, but which are heavier and can’t also be combined with break-action. Naturally all of these guns should be compatible with a bayonet. Also you should be able to make pistol versions of most of these weapons too.
You could have different loads of cartridges too, with the option to use different amounts of poweder, and perhaps loading it purely with modern gunpowder, though that would increase the recoil a lot and only be possible with a reinforced breech.
Oh and firing a black powder charge should produce smoke and make a lot of noise, so firing them indoors a lot would quickly smoke up the room and potentially cause coughing, and it would also attract zombies. You could also perhaps allow the player to make black powder grenades which would have the same effects and be rather limited in their explosive power, but quite easy to make if you have a powder surplus. Stick some in a plastic bottle, bung a string in the top, lob it at a zombie.
But I rather like the idea of making half a dozen flint/matchlock pistols and pulling them all out one after the other to shoot a bunch of zombies, or playing a colonial era big game hunter or something. Muskets are cool and would potentially be quite valuable in a post apocalyptic scenario where machined ammo isn’t possible to make any more.