I've gotten this question a lot -- and it's true, vault qualifying isn't immediately obvious. Here's how it works: A gymnast must opt to do different vaults in the qualifying rounds, and then those two scores are averaged. The eight gymnasts with the highest vault average make it to event finals.
So even though the American women and men competed four gymnasts on vault in prelims, only McKayla Maroney on the women's side and Jake Dalton and Sam Mikulak on the men's side competed two vaults to attempt to qualify into event finals. (And only Maroney and Mikulak succeeded in qualifying; Dalton fell on one of his and didn't qualify).
Why would the other gymnasts opt to only do one? A gymnast must do two very different vaults (from what's called different families, so a Tsukahara and a Yurchenko would be two different families) to qualify into event finals. Many gymnasts struggle to be proficient in two different types of vaults, mostly because with all of the many many things a gymnast must train, adding a second different vault to his or her repertoire is usually down at the end of the list.