There can still be a chemical reaction between zinc metal and copper ions when they are separated from each other. Because the reaction is primarily a transfer of electrons and not a reorganization of covalently bonded atoms, all you need is a path for the electrons to get from the zinc to the copper ions. So, a battery is a chemical reaction in which half of the reaction happens at the anode and half at the cathode.

Below is the same zinc/copper ion reaction you saw above, but separated into two half reactions. The wire provides the path for the electrons to get from the zinc to the copper.