Say I am sampling $x \in \mathbb{R}^2$ from a nice (uniform!) distribution which is supported on the unit disk supported on the origin. Given this is there a way to sample another random vector $y \in \mathbb{R}^2$ such that at least $\mathbb{E} [x_i y_i] = p$ for $i \in \{1,2\}$ for some given constant $p$ and maybe also ensure that the tuples $(x_i,y_i)$ for each $i$ are $i.i.d$?
Feel free to change the distribution of $x$ to something else if that helps you give me such an example!
I am happy to see even examples where just the first condition $\mathbb{E} [x_i y_i] = p$ is satisfied.
For the $x_i$ you gave, $\mathbb{E}[x_i] = 0$, and so $x$ and $y$ being i.i.d. implies that $\mathbb{E}[x_iy_i] = 0$.
If you drop the i.i.d. requirement, then consider $y_i = p/x_i$, which gives $\mathbb{E}[x_i y_i] = p$ (this is a bad idea). A better choice would be $y_i = \frac{p}{\mathbb{E}[x_i^2]} x_i$, which gives $\mathbb{E}[x_iy_i] = p\mathbb{E}[x_i^2]/\mathbb{E}[x_i^2] = p$.
If you want to keep the i.i.d. requirement, then consider $x$ and $y$ sampled uniformly from the disk centered on $(\sqrt{p},\sqrt{p})$. By the iid assumption, $\mathbb{E}[x_iy_i] = \mathbb{E}[x_i]\mathbb{E}[y_i] = \sqrt{p}^2 = p$.