" If you use "appointment," you need an article. "The" appointment if you know which one it is; "an" appointment if it's any appointment. I have no objection to her getting the appointment.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.
means that you have no problem with someone giving her the job.I have no objection to her getting the appointment.
I have no objection to getting her the appointment.