You only need to book if you want to camp in the National Park campgrounds like in Lakefield, at Chili Beach or at Eliot Falls for example. There are loads of free camps all the way up that you can just turn up at, but at that time of year, be prepared to either make camp early to get a decent spot or maybe not all camp right next to each other.
The campgrounds up at the tip get really busy around school holidays too, and although you don't need to book, I personally would if you all want to camp together. If you want powered camping anywhere, you'd probably need to book ahead.
We had a group of 11 adults and 8 kids and I pre-booked several NP campsites for us. We didn't all make it to all of them as some had to drop out, and some were delayed so we did lose the money we'd paid for those bookings, but depending on how tight your budget is, at roughly $5.30?/person/night we didn't complain too much. We also had no trouble with anyone camping in our spot. If you do know far enough in advance that you won't make it on the allocated day and you can get phone reception, you can't get a refund but you may be able to get a camping voucher to use at a later date.
I also pre-booked our accomms in Cairns, Cooktown, and up at Punsand Bay, but only because we had a group and I really wanted to make sure we had sites in some particular places. The rest we just winged it.
Instead of Eliot Falls NP campground, you could free camp up the road at Canal Creek and come the short drive back to swim and use toilets. The McGirr's group did this last year.
The Roadhouses on the way up have good camping out the back and most have a bar, serve meals and have toilets and hot showers for a reasonable price. You don't need to pre-book these if you're just camping.
My kids were 13 and 11 and they loved anywhere that we could swim. Favourites were Cairns Coconut Holiday Resort(BIG4) which was heaven for kids, the Dulhunty River, Eliot Falls/Twin Falls/the Saucepan, Fruit Bat Falls, and Punsand Bay(above ground pool). The Lions Den Hotel was a hit too as you could swim out the back in the river, it had a playground and they had all night pizzas.
Whether you book or not depends on how you all like to travel and camp. I'm a bit of a control freak and stress head so I like to book and have options but still be prepared to be flexible in case of breakdowns etc.... Plus we had 4 kids under 5 years old in our group so I wanted some structure in our itinerary. It would be easier to just wing it and camp wherever when the kids are older.
One thing we did notice though was that tours booked out very quickly before we got up there (lucky I pre-booked some of them too). Our Guurrbi rock art tour and the Thursday Island tour were fully booked out and we tried to book a fishing charter once we were up there but they were all booked out, as were the ones in Weipa.
Good luck planning.
Cheers
Karen