Nonparametric Order Statistics - Does this Exist? I was reading about order statistics on Wikipedia [retrieved 29 June 2022]:

Apparently, if we have a sample with $k$ elements (e.g., $x_1, x_2, ..., x_k$) and assume a probability distribution for each of these $k$ elements - we can also determine the probability distribution for each of these individual $k$ elements.  It seems that the very definition of an order statistic is based on some underlying assumption of a probability distribution.
My Question: Does this mean that for some sample, it is fundamentally impossible to compute the distribution of the $k$th order statistics without some some choice of probability distributions (i.e., non-parametrically)?

The formulae you have cited here are giving you the true distribution of the order statistics in the case where the undrlying sample values are IID random variables from the true distribution $F_X$.  If the form of $F_X$ is unknown then the distribution of the order statistics is likewise unknown, so it is not possible to compute it exactly.  However, is indeed possible to estimate the distribution of the order-statistics non-parametrically.
There are several ways you could go about non-parametric estimation of the distribution of order statistics for an IID sample.  One simple method would be to use a standard non-parametric estimator for $F_X$ (e.g., a kernel density estimator) and then substitute the resulting estimate into the formula for the distribution of the order statistics to yield a "plug-in" estimator.  In this case, you would choose an appropriate kernel CDF/density for your KDE (respectively denoted as $H$ and $h$) and then your estimator for the distribution of the $k$th order statistic would be:
$$\hat{f}_{X_{(k)}}(x)
= \frac{n!}{(k-1)!(n-k)!} \hat{f}_n(x) [\hat{F}_n(x)]^{k-1} [1-\hat{F}_n(x)]^{n-k},$$
where:
$$\begin{align}
\hat{F}_n(x) 
\equiv \frac{1}{n \hat{\lambda}} \sum_{i=1}^n H \Big( \frac{r-x_i}{\hat{\lambda}} \Big)
\quad \quad \quad \quad \quad 
\hat{f}_n(x) 
\equiv \frac{1}{n \hat{\lambda}} \sum_{i=1}^n h \Big( \frac{x-x_i}{\hat{\lambda}} \Big),
\end{align}$$
and $\hat{\lambda}$ is an estimated value for the KDE bandwidth $\lambda$.  This simple "plug-in" estimator does not rely on any parametric assumptions about the true form of $F_X$ and it will give you a locally-consistent estimator for the true distribution of the order statistics.
