Is this the norm when asking for time off?

You're getting the short end of the stick already, by not knowing the usual expectations of a PRN position.

Usually PRN staff inform the facility of their availability. This varies a lot. Some people say "schedule me any days you want, to total 20 hrs a week" (or whatever) and other people say, "I'm available to work on these [specific] days at these [specific] times." The whole point of PRN is that it is a flexibile arrangement in which they don't owe you a set number of hours or any benefits (they only pay you day to day, as needed), and you don't owe them other than, from day to day, agreeing to work for them if they need help and if you are available.

So you are not getting the benefit specifically associated with a PRN position, as you can now see.

It comes down to personal need, preference and choice.

If you are going to be scheduled by someone else according to their preferences, I would look for a position where you also reap the benefits of that (a benefitted PT or FT position). Otherwise, I would forget the vacation this time around and work the hours you're scheduled to work while also turning in your "dates available to work" for the next schedule they are currently working on.

Personally I would not work under the terms/understandings you currently describe. I would give them my available hours and if they can't work with that, I would resign.

Agreed. I once had a PRN job in addition to my FT job. I would let the PRN hospital know what days I am available to be put on the schedule.

They did sometimes call me and ask if I could pick up a shift here and there, and if I could, I sometimes would.

PRN means (usually) you let them know when you are available.

I have never thought the BS of "you need to find your own replacement" was acceptable. No, you are the manager, it is your job to staff the unit, not mine! Especially if someone is calling in sick. How do you want me to find a replacement when I can barely leave my bathroom??? Now, if the schedule is out and posted and I want a day off to see a movie, or whatever, I can understand trying to find someone to switch with me, but not a true call out due to illness (mine or my child, etc).