You will each have an admission ticket. When you get to a FP station, you insert the ticket in, it pops back out, and the FP pops out below (it's also a paper ticket). So, using your family's four admission tickets, you get four separate FPs. Fastpasses move in increments of 5 minutes generally. So, say, 10:35-11:35 would be a FP window, and then the next window is 10:40-11:40. So there's no way your four FP could NOT overlap if you get them at the same time.
One person can carry all four tickets and go get the FPs without the rest of your family.