Good questions. Had to do a little research myself on this.
Salt originates ultimately from the sea. Much of the salt we consume today is mined or evaporated from its deposits. Fresh sea salt is also derived from sea brine using various evaporative techniques.
There's a good bit of info on salt here: http://www.salthealth.org/?content=factsheets
The government recommends 2300 mg/day sodium intake. It is particularly important for replenishing electrolytes in the body. Also, salt iodization
(adding iodine) is an important nutritional consideration--especially for children. So it's important to have not only salt but a source of iodine as well.
In a nutshell--yes, you can "harvest" you own salt in a shtf scenario. In fact, it's been many years since I read "Alas, Babylon," but I seem to recall that the survivors in that story (in Florida) were involved in retrieving salt from a nearby salt deposit.
In Florida, I am sure you would be well situated for collecting salt one way or the other from your environment. Wherever you live in the world, you can do a little geological research to find locations of nearby salt deposits. Ancient seabeds are where they can be found on dry land today. And on seashores, you can process your own saltwater brine to produce salt.