If you go without it being approved, just know that can be viewed as insubordination and ultimately lead to termination. Not trying to scare you, but that is the point of an approval process.
If it were me, I probably would have emailed my boss prior to booking the flight and saying "hey, any update on my vacation request? I am waiting to book a flight and trying to save as much money as possible."
Unfortunately, the only person at fault is you for booking the flight. Not trying to be a cock smack, but I can see HR (that is what I do for a living) telling you that they'll make efforts to accommodate but ultimately, the approval process is there for a reason. Is that a terrible viewpoint for long term retention? Fuck yeah. Does it sound like they care? Nope.
Good luck dudebrohammerstein