Thanks for the date info. At that time of season, you could stay almost anywhere if the four of you will have a vehicle to get around. Mykonos is indeed small, so if you were to stay at or near a beach that has a very limited number of places to eat -- for instance, Elia, Kalafatis, Kalo Livadi, Agia Anna Kalafatis, Panormos, Ftelia -- you would be anywhere from a 10- to 20-minute drive to a variety of restaurants at or near other beaches, in Ano Mera, or possibly even Mykonos Town (or along the roads to those locations).
If you won't have a vehicle but will instead be relying on local transportation like buses, taxis or private transfer services, best beach areas to stay at would be Ornos, Platis Gialos, Psarou, Paraga, Paradise, or Agios Stefanos, where you would be within reasonable walking distance of a selection of good restaurants, or a short bus ride to/from town. Agios Ioannis is another option, though with much fewer places to dine (apart from restaurants situated in most of the luxury hotels in that area).
Just wondering: when you say you're having a hard time finding places to stay, are you referring to limited availability of accommodations on vrbo, or are you referring to hotels?