Is the Čech cohomology of an orbifold isomorphic to its singular cohomology?

As a reference I recommend this paper by K. Behrend:

http://users.ictp.it/~pub_off/lectures/lns019/Behrend/Behrend.pdf

Cech cohomology and De Rham cohomology can be defined for differentiable stacks using a double complex associated to the underlying Lie groupoid. These two cohomologies are isomorphic (see remark 10 p. 262).

Singular cohomology is also defined in this paper (see p. 280), and for real coefficients we have a De Rham isomorphism (see p. 289).

Proposition 36 gives an isomorphism, for any Deligne-Mumford stack $\mathfrak{X}$, between singular cohomology $H^*(\mathfrak{X};\mathbb{Q})$ and the singular cohomology of its coarse moduli space $H^*(\overline{\mathfrak{X}};\mathbb{Q})$.

Example: consider the stack $BG$ where $G$ is a finite group, its coarse moduli space is just a pointy and you recover that rational coholology of $BG$ which is the cohomology of group of $G$ with rational coefficients is trivial.

David C's answer gets this nicely, but another way to intuitively think of it is the following.

Choose a fine enough open cover $\{U_i\}_i$ of $\mathcal{O}$ such that each $U_i$ is isomorphic to $\mathbb{R}^N$ and such that the local groups are all sitting in $O(N)$. Then each $U_i$ and $U_i/G_i$ are contractible, and so all of the terms in your Čech complex are the same.

The maps may vary a little, but over $\mathbb{Q}$ you will get the same result up to isomorphism.

It's a classical result that if $X$ is paracompact and locally contractible, then singular cohomology and Cech cohomology of $X$ coincide, with coefficients in any abelian group. A reference is Spanier's textbook. This applies in particular to the underlying space of an orbifold, in which case a small neighborhood of any point is the cone on the link of the point.