The 12th Amendment linked the two offices on the ballot. A presidential candidate could choose, and a party could nominate a VP of a different party and, if so elected and ratified, take office as a split ticket. The primary process being as drawn out and somewhat more publicly significant makes it hard to pull off something like that. No party wants to cede that power and prestige to the other side. Understandably, if you operate from the notion that your philosophy is the best one for the country. Works a bit different for many state executive offices, where governor and lieutenant governor can be elected separately. That's a lot easier to make a party split.

Other than that, it would take an electoral college tie. House would choose the President, Senate would choose the Vice President. If they're of different parties, you could end up with that.

It was more possible to have a split in the formative and transitional stages of US history as parties were young, changing, and still moving from a parliamentary coalition-building model. The current parties have cemented monetary, identity, and operational holds within the process that would preclude the idea for now. Party hierarchy put the kibosh on McCain selecting Lieberman, for example, and that was probably the closest we've gotten to that possibility in quite a while (to my/public knowledge, perhaps I'm ignorant of other ones or they didn't see much light of day).