I am looking for a Bayesian technique to draw samples from a uniform distribution over the range of a non-invertible (that is, there isn’t even a formula) function $\mathbf{f}: \mathbb{R}^N \rightarrow \mathbb{R}^3$, where $N$ is small and I know the approximate limits for every dimension for both $\mathbb{R}^N$ and $\mathbb{R}^3$. You can assume $\mathbf{f}$ to be differentiable (e.g. using finite differences).
I am failing to see how to use rejection sampling or MCMC for this directly (though there is a related question with rejection sampling as the answer). Is there some way to sample $\mathbb{R}^N$ and reject samples in a principled way to ensure that the retained $(x \in \mathbb{R}^N, y = \mathbf{f}(x) \in \mathbb{R}^3)$ samples look as if the $y$'s were uniformly sampled from $\mathbb{R}^3$?