Using solubility guidelines to predict a reaction of two ionic, water-soluble salts
Of course $\ce{NH4+}$ can join $\ce{Cl-}$ and $\ce{Zn^2+}$ can join $\ce{SO4^2-}$, but the formed salts are completely soluble in water, and they completely dissociate in water to gives the corresponding ions. 

On the other hand, the starting salts:  $\ce{(NH4)2SO4}$ and  $\ce{ZnCl2}$ are completely soluble in water, and they completely dissociate in water to gives the corresponding ions. 

So, all of the ions remain in solution, as if no reaction occurs.