I can't remember Delta's exact policy, but if I recall correctly, you can call ahead of time but when it comes to flight day, they can still refuse you if too many pets have already checked in. Don't quote me on that though.
That's how it was before Avvinue. We just launched the world's first pet travel booking platform so that you can get your pet reserved 100% online and confirms the place on the flight so you don't have sit on hold for hours just to be denied.