AFAIK, Einstein was the first person to propose that c is a universal constant with wider implications than just the speed of light. Special relativity is based on the assumption that the speed of light is constant for every non-accelerating observer - IOW, it never makes any difference how fast you are moving or in what direction; if you measure the speed of light, it’s always the same.
I think it’s just one of those constants that we don’t actually have an explanation for.
AFAIK it’s observationally determined, in the historical sense.
However, we put a lot of confidence in its being a fundamental constant, and so wound up deciding to use c, and our definition of the second, to create the definition for length, the meter. So now c is considered by definition to have its present, exact value, and the second is considered by definition to have its present, exact value, and the meter is c divided by a certain declared time. It’s no longer observationally determined, in this sense.
I believe the speed of light in a vacuum, c, can be derived from Maxwell’s Equations concerning electric and magnetic fields, developed in the 19th century.
Also, I think it is a shame that the popular media have promulgated the idea that the speed of light as a fundamental limit in the universe is analagous to the “sound barrier,” and that only time and technology are required to break the “light barrier.”
It works out that the physical constant, c, is fundamental to many branches of physics [quantum mechanics, gravity, et cetera], and is not confined solely to “the speed of light in a vacuum.”
c = 1, so you might as well ask if the value of 1 is experimentally determined or theoretical. The only reason it might appear to have a value other than one is because humans, for some reason, insist on using different units to measure timelike distances than we use for spacelike distances. Saying that c = 299792458 m/s is basically just the same as saying that 299792458 meters = 1 second, and that statement is really in the same category as saying that 2.54 centimeters = 1 inch.