I think you overestimate how often they use firearms, and exactly how much ammunition they need.
There is very little information on this in-universe so we must apply logic.
On firearm usage
Numerous times throughout the show's run you hear Rick and others talk about conserving ammunition, using melee attacks (i.e. blades), and other forms of creative fighting to avoid using ammunition (e.g. luring them into the ravine in season 5).
Very few times do they use automatic or burst fire. The only times I remember this occurring were when the protagonists were being overrun: Shane during the school supply run, and Rick and Carl during the prison break-in.
Most often we see the protagonists using a revolver (Rick) and basic semiautomatic pistols or rifles (everyone else). Please be aware that not all rifles are automatic: in fact, most are not. Automatic weapons generate a lot of heat: an automatic rifle can actually deform its barrel if fired too much. This is why we have machine guns which are designed for the stress and heat of automatic fire, and I do not recall machine guns being used by Rick's group in the show. Just because you see a long arm in the show does not mean it is automatic: it is likely a semiautomatic rifle or shotgun.
The Walking Dead takes place in rural U.S.A., an area teeming with ammunition. But where is this ammunition?
The basic supply chain is ammunition is built in factories, shipped to gun stores and law enforcement offices, then sold or issued to individual users.
During the impending apocalypse, it is reasonable to assume that people would stockpile ammunition and other supplies. It is natural to assume that as society falls apart, we would all need to defend our little islands of sanity from walkers and other living who want to take our stuff to survive.
So where is the ammunition? As supply chains stop, the bulk of the ammunition would be at gun stores and law enforcement offices. During civilization's final death throes, these would be dried out by survivors looting and pillaging what they can. Most ammunition would likely be in private caches in people's homes.
Throughout the show, one common theme we see is the protagonists looting almost every house they come across looking for food, potable water, guns and ammunition, and other supplies such as baby formula. The show glosses over many mundane aspects of daily life, leading us to assume that unless otherwise mentioned, they have enough of the supplies they need.
Also do not underestimate the fact that the biggest threat is the other living, and they are armed: every living group that Rick's group kills is another opportunity to loot ammunition and other supplies.
On creating new ammunition
During season 6, we see Eugene concoct plans to manufacture new ammunition. Once a group is large and stable enough, this may be feasible. They would need:
Spent casings. Anyone using firearms would need to pick them up off the floor (semiautomatic), potentially during times when they are focused on survival (i.e. not being eaten). This would be easier with revolvers such as Rick's, because they do not eject spent casings automatically.
A supply of lead (Eugene mentions this). Perhaps there are ingots laying around in abandoned factories or foundries, perhaps it would need to be mined (I am not walking into a dark abandoned mine during the zombie apocalypse!). It would not take much: a few hundred pounds of lead is reasonable to achieve in an ingot or two, and individual bullets are tiny.
A foundry. Basically, a facility that can generate enough heat to melt lead, 621.5°F. Your typical wood or propane fire probably cannot get that hot, but a foundry will have the proper tools to generate the heat assuming there is fuel. Lead actually has a low melting point compared to other metals (e.g. Iron melts at 2,800°F) which is a positive thing when civilization grinds to a halt and people want to make bullets.
Someone with metallurgy skills. Dealing with molten lead, pouring it into forms, etc. safely requires specialized knowledge.
If someone such as Eugene has metallurgy skills, comes across an ammunition factory, the factory has raw materials, is not pillaged or otherwise damaged such that it cannot operate, it would be an ideal scenario. Anything less would be a huge amount of work, perhaps insurmountably so. I believe the key is the raw materials: if people need to mine the lead that is a non-starter. As long as there are ingots (preferably) or even mined ore (needs smelting and perhaps mechanical separation of ore and rock), that is a huge boost.
With the living population vastly reduced, there is likely plenty of ammunition for the remaining survivors. Ammunition is stored in places Rick's group is likely to loot anyway, making it easy to replenish as they go along: there is no need to stockpile thousands of rounds of various calibers. Since they kill the majority of the walkers using bladed and other melee weapons, they do not have a need for stockpiling (and carrying) tons of ammunition.
They use guns when should use melee attack (e.g. have few walkers or should avoid noise)They regularly use melee attacks in that situation...