Is it possible to use tap water to make these solutions in the iodine clock reaction?

Other ions in tap water (like Ca²⁺, Mg²⁺, Cl⁻, F⁻) do not affect the result of iodine clock reaction.

The chemicals involved in clock reaction are an oxidizing agent to oxidize I⁻ ions to I₂ (e.g. hydrogen peroxide), a fixed amount of Na₂S₂O₃ to remove a fixed amount of I₂ formed, and starch indicator. All of the chemicals used in iodine reaction do not react with the mentioned ions in tap water.

However, tap water contains a small amount of chlorine (Cl₂) which will oxidize Na₂S₂O₃. If the concentration of chlorine is constant, a fixed amount of Na₂S₂O₃ will be oxidized and thus the amount of Na₂S₂O₃ is still fixed in all runs. Conclusively, the use of tap water does not interfere with the clock reaction provided tap water is used as the solvents of all solutions used.

Chemistry is an experimental science. Try it and find out. (Although distilled water is not expensive and is certain to work.)