Your app is a uifigure, so you can test this outside of AppDesigner. It turns out you need to be fairly specific with your handles, but the code below should allow you to use a horizontal zoom in a uifigure (and therefore in an AppDesigner app).
f=uifigure;
ax=axes(f);
plot(1:10,'.','parent',ax)
h=zoom(f);
h.Motion = 'horizontal';
h.Enable = 'on';
