In fact, for any noncompact Riemannian manifold $M$ (not necessarily with bounded geometry) and for any distribution $f$, there is a distribution $u$ such that $Du=f$, where $D$ is an elliptic operator satisfying the unique continuation property.

This was proved by Malgrange in `Existence et approximation des solutions des ́equations aux d ́eriv ́ees partielles et des ́equations de convolution'. The Euclidean case was discussed in Hormander's book`

Linear partial differential operators' (Chapter 4 as I remembered), and the arguments therein carries over to general Riemannian manifolds.

There is also a constructive proof due to Li and Tam, which is easier to read for geometers.