More out of curiosity than anything else. I know you can always generate a list of random numbers and then sort it, but I was wondering if there exists a (pseudo)random number generator whose output is already in sorted order? I found this, but it and everything else I found only generates integer lists. Is there an equivalent for floats and without worrying about repetition?
You haven't answered the question about what distribution, so I will assume you want the uniform distribution on [0,1].
Yes, this can be done. However, I doubt that the result will be better in practice than just generating $n$ numbers uniformly at random and sorting them.
Based on the order statistics of the uniform distribution, the distribution of the smallest of $n$ numbers from the uniform distribution is known: it has a Beta$(1,n)$ distribution. Also, it is known how to sample from such a distribution.
So, there is a simple algorithm: sample the first number $x_1$ from the sorted sequence, by generating a random sample from Beta$(1,n)$. Then, sample the second number $x_2$, by letting $x_2 = x_1 + d_2/(1-x_1)$ where $d_2$ is sampled from Beta$(1,n-1)$, and $x_3 = x_2 + d_3/(1-x_1-x_2)$ where $d_3$ is sampled from Beta$(1,n-2)$, and so on.
Why does this work? Well, obviously $x_1$ has the right distribution. Also, the numbers $x_2,\dots,x_n$ can be obtained by sampling uniformly at random from $[x_1,1]$ and sorting, so we can generate them recursively using the same algorithm. Finally, unwinding the recursion gives the above algorithm.