Not delivering mail in a timely manner. Am in the right to be upset?

Yes you have a right to be upset.
I am surprised they do not have "mail boxes" for residents.
I would complain to the Social Worker and I would also make a complaint to the State (they inspect facilities) and one final complaint..the Post Office.

Thanks for the schooling, guys. I wasn’t aware that someone had to be approved to deliver the mail. I guess volunteer idea won’t work unless they’re willing to get this certification. Sorry, jph041.

Shad, it sounds as if the attitude is "only authorised people are allowed to sort and distribute the mail. Oh dear, we don't have enough authorised people. Sorrreeee."

It makes sense that the mail must be handled by people who are accountable, trustworthy and competent; it makes sense that they don't want a free-for-all and the resulting mess when items go missing; but it makes NO sense that they haven't got together a reliable routine for a very easily-defined, predictable daily(ish) task.

I think the OP is absolutely in the right to be upset and should feel free to bite someone (metaphorically, of course). This is bullswotsit.

JPH, is there a residents' association or similar consultative group? You want a list of the authorised individuals, you want a weekly timetable, and you want one person per day given ownership of this particular task. You're more likely to get those if you lean en masse on the managers.

Not everyone can deliver the mail. There is some sort of certification and qualification required.

You are right to be upset. Call the ombudsman that monitors this facility and it's residents. They are with the Area Agency on Aging.

Can't residents that are able go down and get their own mail/packages?

Yes, you have a right to be upset. Ask for the social worker to come to your room and tell him/her that your mail is being withheld from you. Suggest that they institute a volunteer program, like Candy Stripers in a hospital, for someone to come in and deliver mail to residents.