Concerning orbifolds there are a lot of misunderstandings. The original definition is due to Ishiro Satake in two papers:

[Sat1] On a generalization of the notion of manifold, Proceedings of the National Academy of Sciences 42 (1956), 359–363. 

[Sat2] The Gauss-Bonnet theorem for V-manifolds, Journal of the Mathematical Society of Japan, Vol. 9., No. 4 (1957), 464–492.

After this papers Thurston decided to change the (not very sexy) name of *V-Manifold* for (the more sexy) **Orbifolds**, and he got some success. Fact is that the notion of orbifold is still, with Satake and Thurston, a space equipped with a smooth structure: Thurston didn't change original Satake definition, he just changed the name.

As spaces with a smooth structure, these orbifolds [are naturally integrated in the category of diffeological spaces][IKZ] and inherit that way all the differential environment.  

Later, the concept of orbifold has changed, and has been associated with a groupoid defining in some sense the underlying orbifold structure. [It is like if you wanted to remind the structure of $S^3$ in the smooth structure of $S^2$ because of the Hopf fibration.] That is the direction taken by Haefliger, Moerdijk and his school. Of course then, the various notions of diffeomorphism, homotopy etc. diverge. The notion of orbifold changed (or refine) again then with the apparition of stacks (but here I'm not familiar enough to have an opinion). 

**But**: diffeologically speaking, if you want to isolate the *internal structure* of the diffeological orbifold you can consider its **structural groupoid** (the germs of the automorphisms of an admissible generating family). Therefore, you can recover what people consider to be the homotopy of the orbifold as the isotropy groups of this structural groupoid. For example, the cone orbifold ${\cal Q}_m = {\bf C}/({\bf Z}/m{\bf Z})$ is clearly contractible, since the retraction $(t,z) \mapsto tz$ is ${\rm SO}(2)$-equivariant, therefore its homotopy is trivial but the structural groupoid has ${\bf Z}/m{\bf Z}$ as isotropy group at the origin and $\{{\bf Id}\}$ elsewhere, that is the information you were looking for. It is not contained in the homotopy group but in the structural groupoid.

Now, it's up to you to choose which direction fits more your needs.

------ Edit March 2017

Coming back to the heart of the question, the legitimacy of this question on morphisms between orbifolds comes from Satake's construction in [Sat2, p.469], where he said, and I cite: 

<blockquote>
The notion of $C^\infty$-map thus defined is inconvenient in the point that a composite of two $C^\infty$-maps defined in a different choice of defining families is not always a $C^\infty$ map. 
</blockquote>

That's because of the required property of equivariance of the local lifting of maps between orbifolds. This problem disapears in Diffeology. Indeed, <b>there exist smooth maps between orbifolds (as diffeologies) that have no local equivariant liftings at all</b>. This is shown in the example 25 of our paper on orbifolds [IKZ]. The function $f \colon \mathbf{C} \to \mathbf{C}$ defines by projectioon a well-defined smooth map between $\mathcal{Q} =\mathbf{C}/\mathbf{Z}_m$ to $\mathcal{Q}_n$ that has no equivarian local liftings on the neighborhood of $0$, any small you take the neighborhood:

  $$
    f(x,y) = \begin{cases}
    0 & \text{ if } r > 1 \text{ or } r = 0 \\
    e^{-1/r} \rho_n(r) (r,0) & \text{ if } \frac{1}{n+1} < r \leq \frac{1}{n}
    \text{ and $n$ is even } \\
    e^{-1/r} \rho_n(r) (x,y) & \text{ if } \frac{1}{n+1} < r \leq \frac{1}{n}
    \text{ and $n$ is odd},
    \end{cases}
    $$

Here $r = \sqrt{x^2+y^2}$, $z=x+iy \in \mathbf{C}$, and $\rho_n$ is a function vanishing flatly outside the interval $]1/(n+1),1/n[$ and not inside.

What is interesting is that, a contrario, a local diffeomorphism between orbifold has always a local equivariant lifting, in any local representation. This is the Lemma 21 of [YKZ].

So, maybe now, I made the point clearer, at least considering the diffeology point of view on orbifolds.


----------------------
  [IKZ] Yael Karshon, Patrick Iglesias(-Zemmour), Moshe Zadka. Orbifolds as Diffeologies. Trans. Amer. Math. Soc. 362 (2010), 2811-2831 
http://math.huji.ac.il/~piz/documents/OAD.pdf

[Sat1] Ishiro Satake. On a generalization of the notion of manifold, Proceedings of the National Academy of Sciences 42 (1956), 359–363. 

[Sat2] Ishiro Satake. The Gauss-Bonnet theorem for V-manifolds, Journal of the Mathematical Society of Japan, Vol. 9., No. 4 (1957), 464–492.