Can the constancy of the speed of light be derived from a deeper theory?
You can't derive it from a "deeper theory", but yes, you can understand why this postulate holds good.
Einstein's second postulate of special relativity is: The speed of light takes the same value ($c$) for any observer in an inertial frame of reference.
Yes. The Michelson-Morley experiment came up with a null result, so Einstein got his thinking cap on and set some groundrules, and the result was special relativity. There was a bit of an issue in that he didn't acknowledge some of the other relativity guys such a Voigt, Fitzgerald, Lorentz, Poincare, but such is life.
I know there is a lot of experimental evidence for this statement, but has it every been proven (theoretically) to be the result of another ("deeper"?) theory, that does not take it as a postulate?
Like I was saying, it hasn't been proven to be the result of a different theory, but IMHO a lot of relativists know that it's because of the wave nature of matter. See http://arxiv.org/abs/0705.4507 by Magueijo and Moffat where you can read this:
"Following Ellis, let us first consider c as the speed of the photon. Can c vary? Could such a variation be measured? As correctly pointed out by Ellis, within the current protocol for measuring time and space the answer is no. The unit of time is defined by an oscillating system or the frequency of an atomic transition, and the unit of space is defined in terms of the distance travelled by light in the unit of time. We therefore have a situation akin to saying that the speed of light is “one light-year per year”, i.e. its constancy has become a tautology or a definition".
The local speed of light is always measured to be the same because we use the local motion of light to define the second and the metre, and we then use them to measure the local speed of light. It's circular, it's a tautology. Note however that whilst the local speed of light is always measured to be the same, a gravitational field is a place where the speed of light is spatially variable. Einstein said this repeatedly from 1908, but there are some cargo-cult issues in that not many people know about it. This example is from 1920, see the second paragraph:
There's other examples from other years, such as this one from 1915: ""the writer of these lines is of the opinion that the theory of relativity is still in need of generalization, in the sense that the principle of the constancy of the velocity of light is to be abandoned". There's also examples from authors such as Shapiro:
Also see this PhysicsFAQ article by editor Don Koks:
"Einstein talked about the speed of light changing in his new theory. In the English translation of his 1920 book "Relativity: the special and general theory" he wrote: "according to the general theory of relativity, the law of the constancy of the velocity [Einstein clearly means speed here, since velocity (a vector) is not in keeping with the rest of his sentence] of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity [...] cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity [speed] of propagation of light varies with position." This difference in speeds is precisely that referred to above by ceiling and floor observers."
I rather think Robert Close explains the deeper meaning rather well in The Other meaning of Special Relativity. Note that it's not some other deeper theory, it's a matter of understanding the evidence of things like pair production and electron diffraction. It's a matter of understanding why special relativity works. it's because of the wave nature of matter. When you and your rods and clocks are made out of waves, you always measure the local wave speed to be the same, because you calibrate those rods and clocks using those waves.