Can each family choose more than one week? Dividing the summer into weeks will work well. Got to figure out what the transition day will be, like “depart by 1 pm Sunday, arrive after 1 Sunday”.
Put a number on a slip of paper for each family, and then each family draws a number out of a hat (or a paper bag) and gets the order in which it chooses. For instance, five families, put slips numbered from one to five. First to draw a number may pick 4, the second might pick 2, the third 1, etc. That’s the order they choose in.
If there is enough room to choose a second week, reverse the original order, so the picks go 1,2,3,4,5,5,4,3,2,1.
And who ever drew first this year will automatically draw last next year, but the other families will draw the order out of a hat again next year.