Just thinking on my feet: From what I can tell, the trouble with setting up a "singular cell" version of homology is that simplices, whether part of a simplicial complex or on their own, have boundaries that are themselves simplices. So the boundary of a singular simplex is a singular complex simply by restricting. On the other hand, the boundary formula for a cell in a cell complex is much more complicated, and taking the boundary of a (mapping of an) abstract cell doesn't continue to carry the same sort of information that the boundary of a (singular) simplex does. In other words, the same thing that let's us glue together CW complexes in ways that are more complex than what we do for simplicial complexes is a hindrance here because we don't have enough extra structure to say that the boundary of a cell is a cellular object in a nice way. I suppose one could try to impose extra conditions, like prescribing that we describe the boundary of a cell in a certain way and then declaring that the singular cellular boundary be made of the kinds of pieces that we get, but now we're well on the road to just recreating singular simplicial homology, though perhaps with some shapes that are a bit different from the standard generalized tetrahedra. This, I believe, can be done: there's such a thing as cubical homology. I don't know much about it, but the basic idea is one uses n-cubes instead of n-tetrahedra. If I remember correctly, as one might imagine, it works out about the same as the tetrahedral version and in its generalized form leads to a notion of cubical sets instead of simplicial sets. My understanding is that each has its technical advantanges, depending what you're trying to do. For example, arguments about products (including homotopies) are going to work more nicely, but then once you've swept that difficulty under the rug, some other lump pops up somewhere else to make something else more difficult (perhaps the simplicial set people will chime in and tell us what).
Sorry that this is more philosophizing than serious answer, but perhaps it will provide some ideas about why we don't generally see the kind of thing you're asking about.