Can amorphous solids have energy bands?

As you can see from this paper (just an example among many), it is perfectly possible to speak about band structure even in amorphous or liquid systems. In a way the problem is simply matter of a sloppy terminology: the meaning of bands of energy was the bare fact that only some intervals of energy were allowed in a solid, and the easiest solids to be explored were the crystalline solids where bands are further decomposed into k-dependent eigenvalues. Actually, the correct way of naming things should be:

• bands are the allowed interval of energy (in some cases such intervals may be overlapping;
• the k-structure of a band in a crystal should be referred to as the dispersion of the band

In classical textbooks like Ascroft&Mermin, you'll find the Bloch wavefunctions labeled with the Bloch wavevector and an integer which is the band index.