I am planning a family trip in July for 5 people ( wife and I plus 3 teenagers). I was thinking 3 nights in Monterosso al Mare then 5 nights in Florence with a few day trips. I know I am a bit late in planning but working diligently to make it work. Any suggestions would be much appreciated.
For that number of people and that number of nights I'd look into an apartment with VRBO or AirBnB, otherwise you'll probably need at least 2 hotel rooms, which will certainly cost at least 150€ each per night at that time of the year. A two bdrm apartment will probably cost you similarly, but will give you a more comfortable stay for a large group. I checked on VRBO and there are still plenty of availabilities in Florence.
For Monterosso I didn't find much at this stage of the game. You are going in the busiest time possible, so you might have to settle for a nearby location, like Levanto.
Just to echo Roberto's advice, you're unlikely to find much in Monterosso at this late date so I'd look in Levanto. What you DO find could be very expensive or at inconvenient locations for accessing the villages. Levanto is larger so should have more rooms/apartments available, and it's only 4-6 minutes from Monterosso by train.