I see, so it just boils down to the fact that I can pick some CW structure on a manifold with corners. Do I need to show that the map $X \to Y$ takes dimension $d$ faces to dimension $d$ faces?

I confess I did not entirely understand where $U$ was coming from when I wrote the question, but now I do. I can show there is a neighborhood of $x$ in the manifold with corners $Y$ which is of the form [open set in interior of $\mathbb R^k$ ] $\times$ [neighborhood of 0 in $\mathbb R^l$] U is the open set in the interior of $\mathbb R^k$.