What is the purpose of the electrolyte in the half-cell where oxidation is taking place?
The primary role of the anode chamber electrolyte (the anolyte) in this case is to provide sufficient conductivity in that chamber.

If deionized water were used in the anode chamber, the conductivity of that liquid would be extremely low. Thus, it's important for there to be something dissolved there in order for any appreciable amount of current to pass through the overall electrochemical cell. The overall electrochemical circuit involves both electronic conduction (in the wires and electrodes) and ionic conduction (in the two electrolytes), and it will not function efficiently/effectively if any element of that circuit is too poorly conductive.

That said: It's not crucial that it be specifically $\ce{ZnSO4}$, however.  $\ce{NaCl}$, $\ce{H2SO4}$, $\ce{NH4NO3}$, or a wide variety of other chemicals might work just as well (or even better!).  However, a sulfate salt makes sense here because (at least):


It provides relatively high conductivity
Its salts with metal cations, including $\ce{Zn^{2+}}$, are relatively water-soluble.
In this specific cell configuration, using sulfate makes sense because it's the same anion as in the $\ce{CuSO4}$ used for the catholyte, and thus you don't have to worry about counter-diffusion of different anions across the salt bridge.


It further makes sense to use $\ce{ZnSO4}$ specifically as the anolyte salt because it provides the same cation as you'll be forming from your anode reaction. This will result in a more stable electrochemical cell, because having a different, potentially electrochemically-active cation in the anolyte would most likely change both the thermodynamic and the practical operating characteristics of the cell.