Why do chloride ions have a smaller ionic radius than sodium ions?

They do not. Sodium ions have a smaller ionic radius than chloride ions. This is because with the chloride ion, the 17 protons have more difficulty attracting 18 electrons in the Cl- ion than sodium does with 11 protons and only 10 electrons to attract with the Na+ ion. With sodium there is 1/11= 10% more attraction (by excess) than in neutral state and hence a smaller ionic radius. With chlorine 1/17= 5.5% less attraction by proton 'shortage' and hence a larger ionic radius.

At the related link there are diagrams of sodium chloride showing the larger size of the chloride ion.