There is a ambiguity about orbifold homotopy, depending of the category you consider. Regarded as a diffeological space — see Orbifold as Diffeology — and considering the general homotopy theory of diffeological spaces. This will give you, for the classical example ${\cal O}_m = {\bf R}^2/{\bf Z}_m$, the trivial homotopy, since ${\cal O}_m$ is smoothly contractible. But you can get a refinement by looking at the principal orbit of ${\rm Diff}({\cal O}_m)$ acting on ${\cal O}_m$ itself, which is ${\cal O}_m - \{0\}$, and you get ${\bf Z}_m$ that is the "good" homotopy of ${\cal O}_m$, according to Haefliger.
So, contrarily {0\}$. Contrarily to manifolds where the group of diffeomorphisms is transitive, the lack of transitivity of the group of diffeomorphisms for orbifolds (the structure groups of an orbifold are diffeological invariants) gives you a family of homotopy groups in each degree, one for each orbit, and especially for the principal orbit. I (still) didn't try to prove that this don't really looked how the homotopy for diffeological orbifolds, coincides with Haefliger's definitionor the homotopy of the orbits of the group of diffeomorphisms, but it's possible. Anywayand the structure groups of the orbifolds (here ${\bf Z}_m$ at the origin) are related, but it has its own logic and it is a diffeological invariant (I consider only what is called "effective orbifolds"). I'm not sure I answer the question, but it may give some directions to dig in.
Note 1: In diffeology that, the singularities of a diffeological space are defined by the orbits of the group of diffeomorphisms, so this homotopy described above for orbifolds is just a specialization of the general case.
Note 2: Here we consider only what is called "effective orbifolds".
