Does a Riemannian manifold have a triangulation with quantitative bounds?

Let us use the following result (e.g., see Eichhorn, Global Analysis on open manfolds, Proposition 1.3)

• If $(M^m,g)$ is of bounded geometry (in the $C^\infty$-sense, say), then there exists $\epsilon_0 > 0$ such that for that for any $0<\epsilon <\epsilon_)$ there is a countable cover of $M$ by geodesic balls $B_{\epsilon}(x_j)$, $\bigcup B_{\epsilon}(x_j) = M$, such that the cover of $M$ by the balls $B_{2\epsilon}(x_j)$ with double radius and the same centers is still uniformly locally finite.

Now choose points $y_k$ such that in each non-empty intersection of $B_{\epsilon}(x_j)$ they span an $m$-simplex, which is diffeomorphic to the standard $m$-simplex by any of the relevant exponential maps $\exp_{x_j}$. Then the maps you are looking for are just Riemannian exponential mappings which satisfy your assumptions by the properties of bounded geometry.

A series of papers has been published (or rather is being published) on this topic, staring with Stability of Delaunay-type structures for manifolds (see http://dl.acm.org/citation.cfm?id=2261284) by Boissonnat, Dyer, and Ghosh. I think the paper in the series you might find most useful is Delaunay triangulation of manifolds (see https://arxiv.org/abs/1311.0117).