Do I need to remove the entire bumper to swap this bracket?

Full bumper removal is the safest approach. You'll need access to the mounting points on both sides, so pulling the bumper off the car makes alignment and fastener work straightforward.

Will this work if I'm replacing a damaged bracket on an existing bumper?

Yes. As long as your bumper is still structurally sound, this bracket bolts in at the factory points. Test-fit the bracket to the bumper before final installation to confirm clearance.

Can I order just the right side, or do I need to buy both sides?

This is the right side bracket only. If your left side bracket is also damaged, order the left side bracket separately.