My siblings did not return the appointment letters to the probate court. I was appointed Personal Representative by the probate courts. Does there not signing paperwork mean it can be interpreted as an objection to the appointment? What legal claims can they make at this point?

Such laws go by state. But given the fact that the court appointed you, that would presume that the time limit to object (as well as apply) had ended. It would take a minute to call your court if you want to make sure.

I agree with Joy.

Not signing what I assume were letters giving consent to your appointment would not constitute an objection to such appointment.