The pentahydrate salt of copper (II) sulfate exists as a blue solid while the anhydrous and monohydrate salts appear as white solids. The structures of these salts are discussed here. The blue colour of the copper (II) sulfate pentahydrate salt arises due to a similar reason why $\ce {Cu^2+}$ in aqueous solution is blue in colour. Based on Crystal Field Theory, this is due to the phenomena of d orbital splitting in the octahedral complex. As discussed in the post linked, in the pentahydrate salt, the $\ce {Cu^2+}$ ion is octahedrally coordinated, with 4 $\ce {H2O}$ molecules forming dative covalent bonds with it while the other two sites are occupied by the oxygen atoms of the sulfate ion. This leads me to wonder if the sulfate ions contribute to any significant degree of splitting of the d orbitals of the $\ce {Cu^2+}$ ion. As the colour of the anhydrous $\ce {CuSO4}$ salt is white, we can infer that the sulfate ions do not contribute to the d orbital splitting. Consequently, we can infer that the sulfate ions do not really form dative covalent bonds with the $\ce {Cu^2+}$ ion.
But why would this be the case? In the sulfate ion, there is rather poor delocalisation of the lone pair due to long $\ce {S-O}$ bond lengths and thus, ineffective p orbital overlap between the p orbitals of the sulfur and oxygen atoms. Hence, the negative charges, as well as the lone pairs of electrons, are rather localised on the oxygen atoms. This analysis all points to the conclusion that the sulfate ion should be able to act as a ligand.
Similarly, the fact that anhydrous, neutral $\ce {CuCO3}$ exists as a gray solid, suggests that the $\ce {Cu^2+}$ ion does not have dative covalent bonds with the carbonate ion , i.e. $\ce {CO3^2-}$ does not act as a ligand. Although for the carbonate ion, due to the stronger delocalisation of electrons within the ion, it may be more convincing to say that it may not be able to act as a ligand effectively.