Why doesn't EDTA complex with alkali metal ions?
If you're going to be picky that a "complex" doesn't form without covalent coordination bonds, then that's your answer. Alkali metals will tend to form cationic salts with EDTA.

Personally, I agree with the other comments. EDTA does form complexes with alkali metals. 

I'd classify the disodium salt of EDTA as a complex. It has a metal and an organic ligand, and there's no direct metal-carbon bond (which would make it organometallic). As discussed in another question, it's a pretty arbitrary distinction between an ionic bond and a covalent one:


  What happens if the electronegativity difference is exactly 2.1?


This distinction is particularly cloudy in the case of metal complexes, since most metals readily gain positive charge.