I'm sure the answer to your question is "it depends on the application". Here are three categories that come to (my idiosyncratic) mind.

Perhaps the most general category in the direction you're looking is a version of Lawvere's category of metric spaces. Recall that $\mathbb R_{\geq 0}$ is a category, on account of the fact that it is a poset: there is a unique morphism $x\to y$ whenever $x\geq y$. It can be given a symmetric monoidal structure by declaring that $\otimes = +$. Then an $(\mathbb R_{\geq 0},+)$-enriched category is nothing but a (generalized) metric space. The natural "functors" are the distance-non-increasing maps.

So I would suggest that a good guess, if you must make a guess, for a category of Riemannian manifolds has as its objects all Riemannian manifolds $(M,g_M)$ (of whatever regularity you like) and its morphisms $f : (M,g_M) \to (N,g_N)$ are smooth maps $M \to N$ such that at each $m\in M$, the symmetric bilinear form $g_M - f^* g_N$ on the tangent fiber at $m$ is positive-semidefinite. This is sort of an "infinitesimal" version of the Lawvere one. Or just notice that every Riemannian manifold gives a metric space, and use the distance-nonincreasing maps (in infinite dimensions this is more subtle, as there are many important examples in which distinct points are connected by arbitrarily short paths).

But here are two other categories "of Riemannian manifolds" that are important in quantum field theory:

In one approach to understanding $n$-dimensional quantum field theory (in Euclidean rather than Lorentzian signature), one constructs a category whose *morphisms* are $n$-dimensional compact Riemannian manifolds with boundary, and whose objects are germs of $n$-dimensional Riemannian manifolds around $(n-1)$-dimensional compact manifolds. For details on this approach, one should start with the work of Stolz and Teichner.

A category I am particularly fond, which provides a "weaker" notion of $n$-dimensional Riemannian quantum field theory, of has as its objects $n$-dimensional open Riemannian manifolds, and its morphisms are isometric embeddings.

And I'm sure that there are other equally interesting categories, especially if you are interested in infinite-dimensional manifolds, which I haven't really even touched.

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