Step 1: Identify the denticity of the ligand. Oxalato, \(C_2O_4^{2-}\), chelates the metal using both of its carboxylate oxygens, so it is bidentate and occupies two coordination sites.
Step 2: Count the ligands. The formula \(K_3[Co(C_2O_4)_3]\) contains three oxalato groups inside the coordination sphere.
Step 3: Multiply sites per ligand by number of ligands: \(2 \times 3 = 6\) metal-ligand bonds.
Step 4: The coordination number equals the number of such bonds, so it is \[ \boxed{6} \] matching option (iv).