If my desired major is not at a school I want to apply at, would I get rejected from that school?

As far as I know, you won't be rejected from a school simply because they don't have your desired major. The only concern I could see is that a college might not think they'd be a good fit for you--which, if your desired major is really important for you to have, maybe you should consider applying to a different school so you can have that match.

But, if the fact that this school doesn't have your desired major hasn't deterred you (which I'm assuming it hasn't), find a similar major(s) which interest you and delve into that whenever/if you're asked to. If you're interested in Public Policy, but your school only has Political Science and Economics, talk about those majors instead. Or, talk about your desired major in your application, and explain how the majors and courses that school offers will help give you the education you desire. Unless your desired major is extremely niche and specific, I think you'll be able to find a mix of different disciplines to match the same education you'd be receiving as if you were studying your desired major.

Is the application a school specific one? If so email the school directly and ask how to proceed. If you for some reason don’t hear back just have it at the top of the individual section. Also for any school supplements (if any is there a way to talk about your major?)