Relation between frequency spectrum and PDF of a random variable
Let's take this from the bottom:


  Is this a reasonable question?


Of course!


  In other words, would the PDF of a random variable constrain the possible frequency spectra the resultant random process may take?


That depends. For example, if the PDF of your RV was a Dirac impulse, i.e. your RV was actually a constant, then, obviously, yes, that will influence spectral aspects of your random process.

Now, in general, a random process is random, i.e. one can not say what its spectrum is going to be (or whether that is possible to define at all).

Hence, you can only do things like considering a specific statistical/stochastic property the random process has. 

That, for example, and also most likely closest to what you're looking for, would be the Power Spectral Density (PSD).

Now, intuitively, one would just say that you could take infinitely many realizations of a random process, transform them to the frequency domain (using the Fourier transform), and calculate the expectation value of these transform's magnitude squares. 

However, as Matt hinted at, it's really not easy to show that this transform, or the expectation value, are well-defined for some types of processes (or if you can't make some strong assumptions on the process' properties).

Hence, mathematically, you take the elegant detour of first calculating the autocorrelation, which, again, is a property of the process, and Fourier transforming that.

For particularly  well-behaved processes (mainly: Your process needs to be wide-sense stationary), this identical to the aforementioned expectation of the magnitude squared Fourier transform across all realizations; that's the Wiener-Chintschin theorem, which I mentioned in another answer that might be of relevance to you.

The point here is: Together with some statements on the autocorrelation of the process, and by observing the PDFs at fixed times, you might come to some statements on the process' PSD.


  When looking at the frequency spectrum of the generated data does the knowledge of the exact distribution function have any constraints/give any information about the frequency spectrum of the samples?


No. This is exactly the case where lack of ergodicity means that you can't derive much from a time ensemble.