Wilderness Tarpon 130s are a good option. They have been discontinued but you should be able to find some used ones at a decent price. A friend has one and uses it both Solo and Tandem and loves it. It seems to track and turn well and is not too heavy. Lots of rigging options. Plan on the bigger kids that will be doing the majority of the paddling sitting in the rear seat and younger children in the front seat will probably just play at paddling but they can learn this way.
Another good one is the Malibu II XL about the same size and weight capacity as the Tarpon, lots of rigging options and tracks really well. It may be a little more stable than the Tarpon. It has the same solo /tandem option as the Tarpon 130. You will all ssuffer the cheaper Yaks as most of them will not track as well and not be as stable as some of the other kayaks. Look for reviews on Paddling.net
You will definitely want a utility trailer of some sort to carry them on as well as some kind of box to carry all the gear in. Get at least 4 good paddles for the Primary Paddlers. A Cheaper paddle can do for the Younger ones. Also fit everyone for a good PFD that they will wear all day. I have a friend who carries 4 kayaks on top of her tent camper on racks. She has 2 of the Malibu IIs for her 4 kids, who not only fish from them, but use them in the lakes as swimming/ diving platforms. Her kids a are 9-14. Her and her husband have OK Tridents. They all have a great time paddling and camping together. Right now they are working their way through all the State Parks in Texas. I'm sure your family will enjoy the new activity as much as they do.
As for inflatables, there is the risk like others said of oyster shells, as well as they will not paddle as well as a plastic hull kayak. It is the same with the cheaper plastic kayaks. The plastic is thin and flexes so much that it affects the performance. Over time, the inflatable boats will stretch also and won't handle well at all.
Good Luck and Good Paddling.