Does noncompact manifold or orbifold have the homotopy type,of CW complex? I forget for a while, we don't need the compactness condition here right?
According to The Topology of CW-Complexes by Lundell and Weingram
(Van Nostrand Reinhold, 1969) the answer is yes for (separable) manifolds.
For smooth manifolds the following holds.
By the existence of Morse functions one can deduce a handle-body decomposition of a manifold.
This decomposition then yields a CW structure on a space homotopy equivalent to the manifold.
I don't think that compactness is needed in any of the above arguments.