Salt is very important to humans, and it always has been throughout history. Its main purpose in medieval times was the preservation of food and supplementing diets. A long time ago people noticed how food would keep for longer when salted and also that people with salt poor diets died sooner. Due to all these important attributes as well as more spiritual ones (such as salt being important in religion and a sign of fertility of marriage) it was an important resource that many vied for control of. I am aware of saltworks at Uffering, Saltshore and Saltpans, and there are probably others that I don’t know about, but even with our scale this is nowhere near enough to a) sustain the population of Westeros and b) fulfil the canon (salt is mentioned A LOT). First I will go through the main methods of collecting salt:
1. Collecting saltwater. This method is one of the most ‘primitive’ and is very old, originating to when humans first learnt ironworking. It involves collecting saltwater and then leaving them outside for the sun to evaporate, or in climates where it is not efficient to do this because of rain and low temperatures heating the objects that store the saltwater. However the second way is not very efficient and early on salt merchants learnt it was far more efficient to harness the power of the sun instead. This method would be particularly suitable in Dorne which has a high temperature and low rainfall but would also work in other regions in the south. In medieval times the furthest north the solar-based approach worked was the south coast of Brittany. I am not sure where that equates to in Westeros though. It is also restricted to areas with salt water, so the coasts.
2. Saltwater pools: A more advanced technique of the previous method where more saltwater is involved and it is trapped in pools and then they wait for it to evaporate. Seems only feasible in areas with strong regional authority who can oversee the process.
3. Multiple Connected pools: The latest advancement in salt technology (around 6th century CE). It had a single pond where water would evaporate and then pumps and sluices that moved the more concentrated salt solution onto the next pond, and then the next, and so on. At the same time new saltwater was funnelled into the first pond in the system so it kept happening. When the saltwater got to a certain salinity further along down the pond system the salt crystallises onto the bottom where it can be scraped off. However this process is very slow and therefore produces coarse rather than fine salt, but coarse salt is still fine for salting meat and fish. It requires not too much equipment and little manpower (except for the end when you have to scrape the salt off the bottom of the ponds). This method is particularly affected by too much rainfall as it dilutes the solution.
4. Rock salt: I will not pretend to be an expert on the formation of salt mountains and why it is where it is but many societies even as early as the Celts in central Europe have mined salt. Sadly I think there aren't any salt mines in Westeros at the moment (I may be wrong about that), but hopefully that will change. The basic method was just normal mining, and taking rocks out the mountain where they would be crushed into salt, but in the 13th century a new method was developed where miners dug out a vein of salt but then water was piped in, and when it become a dense brine it was piped out again and boiled down into crystals. Since this was developed in the Dürnberg mountains, this should only happen in mountain ranges. Maybe the Vale and the mountains in the North would be especially suitable to this given the lack of competition from sea salt facilities.
5. Peat Salt: Not a common method, mostly confined to Holland where rivers meet the sea in estuaries. Sea-logged peat is carried from the river mouths and then it is dried and burnt, leaving ashes and salt. Next saltwater was added which absorbed the salt but left behind ashes. It formed a brine which could then be evaporated. This method could go wrong and make impure “black salt”, or it could produce some very white and finely grained salt if the peat wasn't mixed with soil. Maybe this method could work in the Neck?

Salt and the North: Ironically, the place that needed the most salt was Northern Europe, where it is tricky for salt to be produced cost-effectively. This meant that salt boiling facilities sprang up across Northern Europe despite it being very inefficient compared to Southern Europe. It was needed there for one reason: cod. Cod is from a family of low-fat fish which basically means it is much easier to salt and can be kept very easily compared to some of its more oily cousins, but it is only rich and in dense shoals in richer northern waters. In the 16th century, which correlates quite nicely to the late medieval setting of Game of Thrones, around 60% of fish eaten by Europeans was cod, which is pretty crazy. Therefore I think salt production would occur in the North but only would be worth it next to major fishing areas and where importing salt would be more expensive.

Exporting: Salt can be exported but it is only really efficient to do so in large quantities, or when it is added to fish or meat as value is added per kilogram.

Salt and Government: Many ancient nations on seeing the value of salt would try and either have a state monopoly on it (like China did at some points), or regulate + tax it heavily, like what Rome and most medieval countries did. Both would also subsidise it at a lower price to the poor at points because to be blunt it made workers last longer. I would not put it out of the question that Littlefinger regulated salt heavily given his ways with coin.

All this stuff is from a book I'm reading called Salt (which is actually pretty good)
Thanks for reading