Remark 1. From a pseudogroup $\Gamma$ on $M$ (for example the local symplectomorphisms if $M$ is equipped with a symplectic form, or the local diffeomorphisms preserving a foliation...) an (abstract) groupoid $\mathcal{G}=\mathrm{Germ(\Gamma)}$ can be obtained via the construction of germs, and I believe it's also a (usually infinite dimensional) Lie groupoid. In this case $\mathcal{G}_0$ is some space of germs of manifolds, and $\mathcal{G}_1$ consists of germs of local diffeomorphisms belonging to $\Gamma$.

Remark 1. From a pseudogroup $\Gamma$ on $M$ (for example the local symplectomorphisms if $M$ is equipped with a symplectic form, or the local diffeomorphisms preserving a foliation...) an (abstract) groupoid $\mathcal{G}=\mathrm{Germ(\Gamma)}$ can be obtained via the construction of germs, and I believe it's also a (usually infinite dimensional) Lie groupoid. In this case $\mathcal{G}_0$ is some space of germs of manifolds, and $\mathcal{G}_1$ consists of germs of local diffeomorphisms belonging to $\Gamma$. (Edit: for the benefit of the readers let me say D.Carchedi in his answer below has pointed out that this $\mathcal{G}$ is actually finite dimensional).

(Warning: I'm not an expert in the topic) Let's work in a "geometric" category, for example the category $\mathfrak{Diff}$ of "manifolds" (without the requirements of connectedness and second countability and even of finite dimensionality), and consider also the subcategory of usual manifolds $\mathrm{Diff}$. We can than consider two kinds of objects:

(Warning: I'm not an expert in the topic) Let's work in a "geometric" category, for example the category $\mathfrak{Diff}$ of "manifolds" (without the requirements of connectedness and second countability and even of finite dimensionality), and consider also the subcategory of "usual" finite dimensional (Edit: possibly disconnected, Hausdorff, second countable) manifolds $\mathrm{Diff}$. We can than consider two kinds of objects:

Q3. More generally, given a groupoid $\mathcal{G}=(\mathcal{G}_1 \rightrightarrows \mathcal{G}_0)$, is there a manifold $\mathcal{S}$ in $\mathfrak{Diff}$ such that $\mathcal{G}$ is equivalent to the groupoid $\mathrm{Germ} (\Gamma)$ of germs of some pseudogroup $\Gamma$ of diffeomerphisms of $\mathcal{S}$ ?

Q4. What about other "geometries" (i.e. other sites in which to consider internal groupoids and -assuming the notion can be generalized- pseudogroups)?