The high school diploma is only important in that it means that some organization thought you had learned enough to be worthy to receive it. That said, the college is interested in seeing you graduate—in part because if you don’t, it reflects poorly on them statistically, and in part because if you’re not going to graduate, they’re less interested in admitting you. Getting the piece of paper shows a certain level of commitment and persistence, and those are traits that are incredibly valuable in education.
All that said, some schools might insist on it, but others won’t. Generally, the smaller and more elite the school, the more likely it is to look at non-traditional students.
So there’s no hard and fast rule—some universities will insist on it, while others won’t. And some universities that officially insist on it will make an exception for a promising student, while others that don’t officially insist on it will put an application from someone without a high school diploma at the bottom of a pile.
It’s probably worth finding out if there’s anything you can do to get the high school diploma, if only because it will make things significantly easier and less uncertain.