Converting fair draws from a PDF into draws from a PDF with a reduced dimensionality Suppose I have N fair samples $(x_i, y_i); 1 \leq i \leq N$ drawn from the 2 dimensional continuous PDF $p(x, y)$; $\int p(x, y) \, dx dy = 1$. The functional form of $p(x, y)$ is unknown.
Is there a way to obtain from these samples a set of fair draws (not necessarily of the same number) from the reduced distribution $p(x, y = y_0)$ where $y_0$ is some constant?

No, continuity isn't enough.  With probability 1 you have no samples where $y_i=y_0$, and there will be some interval $[y_l,y_u]$ around $y_0$ where you have no samples.  A continuous function $p(x,y)$ can take any value at $y=y_0$ if there's an interval around $y_0$ where you have no data.
If $p(x,y)$ is not only continuous but you have some finite bound on its derivatives with respect to $y$, then you can approximate draws from $p(x,y_0)$ by taking just the subset of your original samples that have $y_i$ close enough to $y_0$. You can improve this a bit by including samples with a probability based on $y_i-y_0$ rather than just including all samples where $y_i-y_0$ is below a threshold.
You still can't get draws that are known to be exactly from $p(x,y_0)$ for finite $N$.
