Rental Agencies/websites for Cinque Terre

AirBnB?????

As Emily suggested, airbnb is probably your best best. Just on a poke around the net, I ran into a few agencies but have zero firsthand experience with any to recommend or not. If nothing else, they'll give you an idea of what apartments are going for in July ($$$!) and sleeping arrangements (yes, many sleep all 4 in the same room but that's not unusual for US hotel rooms either.) You are very late in the game for booking for high season, and might look at Levanto instead of any of the 5 villages? It's a scant few minutes by train into the region, and may have more offerings than the CT.

In any case, what I will caution you to do is to check the actual location of a rental very carefully as some can be far up in the hills or otherwise some distance outside of the villages, making getting around time-consuming and difficult. You want to be within reasonably easy walking distance from amenities and train stations.

Booking.com lists apartment rentals
VRBO