Can every compact subset of $\mathbb R^n$ be realized as a manifold with boundary? I think it should be true since in Lee's introduction to smooth manifolds, he referred to a differential form on a compact subset $D \subset \mathbb R^n$. It is easy to provide charts for $\operatorname{Int} D$, but for $\partial D$, how should I set up the charts for it?

No. Just look at the Cantor set in $\mathbb R$ as an example. It has no neighborhoods homeomorphic to any Euclidean space (or half space), but it is compact.
For a less pathological example, consider
$$\left\{\frac1n:n\in \mathbb N\right\}\cup\{0\}$$
This is compact, but no neighborhood of $0$ is homeomorphic to Euclidean space or half space.
