And you say there's kids at home who can check the incubator for temps- if they're able to reach you by cell phone or something, you should probably leave them at home. The chick may take a while to hatch, and they don't usually need food for the first 24-48 hours after hatching (they can even be left in the incubator, though I never do, I like to fluff mine in the brooder). The car motion would most likely be bad for those who haven't pipped yet, and it's very hard to control temps and humidity on the go. Aside from that, once they've hatched it's hard to keep them stable on the go, and it will sort of dock your ability to do much at the ocean. Give the kids some responsibility, they have to learn it sometime!
As for the chick food, water, etc, you can take a shallow bowl and put a cleaned stone in the dish, and use that as a water bowl. The stone should keep anyone from drowning in the water- and if you want to, you can cook the stone in the oven at ~250-300 degrees for half an hour and it will sterilize it. Chick feeding bowl can be done the same way.
You can grind up various things for them for a day or two till you can get the actual starter; cheerios, corn meal, oatmeal (not instant), hard boiled eggs, kitten food.... Keep in mind that the chicks probably won't need to eat for the first day (sometimes two) after hatching, so you won't have to worry too much.
