There have been some reports of tickets running out, but in our experience it wasn't an issue. I planned on running to get the Anna and Elsa tickets upon boarding (the location varies on ships and sailings so check the navigator) but got distracted meeting other characters and with my daughter losing her first tooth. By the time I remembered, we'd been on board quite a while, and we weren't even on board early to begin with.
When we got tickets, there was no one else there and they had piles left. We hung around for a min, used the bathroom to clean up, came back and still no one was around, so there didn't seem to be a mad rush and everyone who wanted tickets seemed to get them.